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Ab initio crystal orbital calculations on polyacetylene-polyethylene copolymers
V o l u m e 126, n u m b e r 1
PHYSICS LETTERS A
14 D e c e m b e r 1987
AB INITIO CRYSTAL ORBITAL CALCULATIONS ON POLYACETYLENE-POLYETHYLENE COPOLYMERS M.A. ABDEL-RAOUF, C.-M. LIEGENER and J. LADIK Chair for Theoretical Chemistry of the Friedrich-Alexander-University Erlangen-Nfirnberg, Egerlandstrasse 3, D-8520 Erlangen, FRG Received 21 August 1987; revised manuscript received 15 October 1987; accepted for publication 22 October 1987 Communicated by A.A. Maradudin
T h e b a n d s t r u c t u r e s o f p o l y a c e t y l e n e - p o l y e t h y l e n e c o p o l y m e r s o f v a r i o u s c o m p o s i t i o n s h a v e been calculated. T h e results h a v e been c o m p a r e d to the b a n d s t r u c t u r e s o f p u r e polyacetylene a n d polyethylene, a n d possible i m p l i c a t i o n s for m o l e c u l a r electronic devices h a v e been discussed.
oo
1. Introduction
S(k)=
~ q= - ~ :
Semiconducting superlattices can be synthetized by alternating epitaxial growth of thin layers of two semiconducting components. Quasi-one-dimensional superlattices, i.e. copolymers, have in some cases (although only with random sequences) also been obtained [ 1,2]. Such systems may have several possible technological applications, e.g. as semicondutor diode lasers [ 3], electro-optical modulators [ 4], non-linear optical devices [ 5 ] and molecularelectronic switches [ 6 ]. In the present Letter we report the results of some ab initio crystal orbital calculations on periodic quasione-dimensional polyacetylene-polyethylene copolymers.
2. Description of calculations
The calculations have been performed within the framework of the ab initio crystal orbital method [ 7 ]. This means that one has to solve the matrix equation
F(k)cn(k) =~.(k)S(k)cn(k) ,
(1)
with F(k)--
~. q= -c~o
38
exp(ikqa)F(q),
(2)
exp(ikqa)S(q),
(3)
where e,(k) gives the band structure, F(q) and S(q) are the Fock and overlap matrices, defined by [S(q)]rs = (Zr° IZq) ,
(4)
[F(q)]r~ M
=(ZrOI--LI/2 -
~
~ (Za/lr-Rq'l)lZqs)
ql = - o o a = I
+
~,
~ P(q,-q2)uv[ (ZrZ,, o q, IXIX,+-")
ql ,q2 = --'~ 11,/= 1
! /~,O~,ql
- - 2 \ , g . rAu I x g 2 X " ~ ) ] ,
(5)
where Xq is the rth atomic basis function placed in cell q, m is the number of basis functions in the unit cell, M is the number of atoms per cell and P(ql--q2)uv are the elements of the density matrix for the polymer. All band structure calculations have been performed using the STO-3G minimal basis sets [8] and fixed standard geometries ( C - C = 1.54/~, C - H = 1.09 A, translation distance=2.52 A for polyethylene, C - C = 1.43 A, C=C= 1.36 /k, C - H = 1.08 A, translation distance =2.41 A for polyacetylene). The polymers Ax and Bx, where A=C2H2 and B = C2H4, have been calculated in the second-neighbor interaction approximation. The mixed systems 0375-9601/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 126, number I
PHYSICS LETTERS A
(AB)x and (A2B2)x have been calculated in the firstneighbor interaction approximation. For comparison we have calculated the (AB)x system as well in the second-neighbor interaction approximation. It should be mentioned that the number of atoms explicitly considered in the second-neighbor approximation of (AB)x is nearly equal to the number of atoms considered in the first-neighbor approximation of (A2B2)x.
3. Results and discussion
The resulting band structures are represented in fig. 1. There we find the bands which are situated in the valence and conduction band regions of the pure polymers. In the first column (from the left) are given the bands of (AB)x in the first neighbor approximation (N= 1). The second column contains the resuits for N = 2, which are drawn next to the results of (A:B2)x for N = 1 because of the argument at the end of section 2. (It can be seen that the N = 1, 2 resuits for (AB)x are almost identical, which demonstrates the convergence with the increase of N.) The last two columns contain our results for the valence
En(k) eV) +10
mmd --
d(A)
Tf"
d(A)
_ d(A)
}o
14 December 1987
and conduction bands of polyacetylene and polyethylene (compare with ref. [9]). The states which are mainly localized on the sublattice A (or B) are denoted in the figure by the letter A (or B), d refers to states which are delocalized over both sublattices, and d(A) assigns states which have components in both sublattices, but with considerably larger A-type components. The letter rt indicates the type of symmetry characteristic to the corresponding eigenvectors of the state considered. The width of the narrow (A) valence band of the system (AB)x is 0.0675 eV, while the widths of the narrow valence and conduction bands, (the d(A) bands), of the system (A2B2)x are 0.0564, 0.0606, 0.0375 and 0.0565 eV, respectively. Figs. 2a and 2b contain the details of the band structures of both systems. As can be qualitatively expected from negativefactor-counting calculations in the simple tight-binding approximation [ 10], there occurs in the copolymers a number of relatively narrow subbands, with gaps in between, in the region of the bands of the corresponding pure polymers. These subbands can be assigned to either delocalized (bulk) states which belong to both components of the copolymers, or to states which are localized only on one of the two (A or B) sublattices. It turns out that the highest valence band is of such a sublattice type, namely localized on the acetylene units, and has n symmetry. This is due to the fact that the A-type valence band is reaching to higher energies, than the B-type. Since these sublO, O
J
~
O
f
0,01
0,0'
n:
(AB)x (AB)x (A2B21x N:
1
-'°
__d(A) d(A)
2
1
Ax
10'0 1
E(eV)
E(eVl lO,0-
10,0-
20.0-
20.0-
-20
2
Fig. l. Band structure results for the (AB)x and (A:Bz)x copolymers where A=C2H2 and B=C2H4. Also the pure polymer results A,, B, have been included. The ordinate gives the band energies in eV. The types of the bands are indicated.
0,0
k
1Tlo
0,0
k
TTIo
Fig. 2. (a) Band structure of (AB)x system ( N = 1). (b) Band structure of (A2B2).,-system ( N = 1 ).
39
Volume 126, number 1
PHYSICS LETTERSA
b a n d s correspond to states which can be described by the A-type elementary cells with a translation vector which bridges the B-type cells, the widths of these subbands are very small. In the energy region where the A-type a n d B-type valence b a n d s overlap, wider s u b b a n d s of mixed eigenvector character appear. These s u b b a n d s are separated by a very small gap from each other in the case of (AB),, b u t they are overlapping in the case of (A2B2).,-. These results confirm the picture expected from simple model considerations for copolymers with partially overlapping bands. P s c h e n i c h n o v [ 6 ] has suggested that such copolymers could be used as molecular-electronic switches because the periodic superstructure permits resonant t u n n e l i n g which can be switched o f f b y a local change of the energy levels. O u r ab initio b a n d structure calculations have shown that in the case of copolymers with partially overlapping b a n d s really narrow s u b b a n d s are f o u n d which correspond to an array of potential well states and thus support the validity of the model considerations in a realistic example.
Acknowledgement The financial support by the "Deutsche Forschungsgemeinschaft" (Project-no. Se 463/1-1) a n d
40
14 December 1987
by the " F o n d s der Chemischen Industrie" is gratefully acknowledged.
References [ 1] K. Kanazawa, A.F. Diaz, M.T. Krounbi and G.B. Street, Synth. Metals 4 (1983) 119. [2] O. Ingan~is,B.O. Liedbergand W.U. Chang-ru, Synth. Metals 11 (1985) 239. [ 3] N. Holonyak Jr., R.M. Kolbas, R.D. Dupuis and P.D. Dapkus, IEEE J. Quantum Electron. 16 (1980) 170. [4] D.A.B. Miller, D.S. Chemla, T.C. Damen, A.C. Gossard, W. Wiegmann, T.H. Wood and C.A. Burrus, Appl. Phys. Lett. 45 (1984) 13. [ 5 ] D.S. Chemla, D.A.B. Miller, P.W. Smith, A.C. Gossard and W.Wiegmann,IEEE J. Quantum Electron. 20 (t984) 265. [6] E.A. Pschenichnov, Sov. Phys. Solid State 4 (1962) 819; J.R. Barker, in: Proc. 2nd Int. Workshopon Molecular electronic devices, ed. F.L. Carter (Dekker, New York, 1983). [7] G. Del Re, J. Ladik and G. Bicz6, Phys. Rev. 155 (1967) 977; J.-M. Andr6, L. Gouverneur and G. Leroy, Int. J. Quantum Chem. 1 (1967) 427, 451. [8 ] W.J. Hehre, R.F. Stewart and J.A. Pople, J. Chem. Phys. 51 (1969) 2657. [9] J. Ladik, J.-M. Andr6 and M. Seel, Quantum chemistry of polymers - solid state aspects ( Reidel, Dordrecht, 1984). [ 10] M. Seel, Chem. Phys. 43 (1979) 103.
PHYSICS LETTERS A
14 D e c e m b e r 1987
AB INITIO CRYSTAL ORBITAL CALCULATIONS ON POLYACETYLENE-POLYETHYLENE COPOLYMERS M.A. ABDEL-RAOUF, C.-M. LIEGENER and J. LADIK Chair for Theoretical Chemistry of the Friedrich-Alexander-University Erlangen-Nfirnberg, Egerlandstrasse 3, D-8520 Erlangen, FRG Received 21 August 1987; revised manuscript received 15 October 1987; accepted for publication 22 October 1987 Communicated by A.A. Maradudin
T h e b a n d s t r u c t u r e s o f p o l y a c e t y l e n e - p o l y e t h y l e n e c o p o l y m e r s o f v a r i o u s c o m p o s i t i o n s h a v e been calculated. T h e results h a v e been c o m p a r e d to the b a n d s t r u c t u r e s o f p u r e polyacetylene a n d polyethylene, a n d possible i m p l i c a t i o n s for m o l e c u l a r electronic devices h a v e been discussed.
oo
1. Introduction
S(k)=
~ q= - ~ :
Semiconducting superlattices can be synthetized by alternating epitaxial growth of thin layers of two semiconducting components. Quasi-one-dimensional superlattices, i.e. copolymers, have in some cases (although only with random sequences) also been obtained [ 1,2]. Such systems may have several possible technological applications, e.g. as semicondutor diode lasers [ 3], electro-optical modulators [ 4], non-linear optical devices [ 5 ] and molecularelectronic switches [ 6 ]. In the present Letter we report the results of some ab initio crystal orbital calculations on periodic quasione-dimensional polyacetylene-polyethylene copolymers.
2. Description of calculations
The calculations have been performed within the framework of the ab initio crystal orbital method [ 7 ]. This means that one has to solve the matrix equation
F(k)cn(k) =~.(k)S(k)cn(k) ,
(1)
with F(k)--
~. q= -c~o
38
exp(ikqa)F(q),
(2)
exp(ikqa)S(q),
(3)
where e,(k) gives the band structure, F(q) and S(q) are the Fock and overlap matrices, defined by [S(q)]rs = (Zr° IZq) ,
(4)
[F(q)]r~ M
=(ZrOI--LI/2 -
~
~ (Za/lr-Rq'l)lZqs)
ql = - o o a = I
+
~,
~ P(q,-q2)uv[ (ZrZ,, o q, IXIX,+-")
ql ,q2 = --'~ 11,/= 1
! /~,O~,ql
- - 2 \ , g . rAu I x g 2 X " ~ ) ] ,
(5)
where Xq is the rth atomic basis function placed in cell q, m is the number of basis functions in the unit cell, M is the number of atoms per cell and P(ql--q2)uv are the elements of the density matrix for the polymer. All band structure calculations have been performed using the STO-3G minimal basis sets [8] and fixed standard geometries ( C - C = 1.54/~, C - H = 1.09 A, translation distance=2.52 A for polyethylene, C - C = 1.43 A, C=C= 1.36 /k, C - H = 1.08 A, translation distance =2.41 A for polyacetylene). The polymers Ax and Bx, where A=C2H2 and B = C2H4, have been calculated in the second-neighbor interaction approximation. The mixed systems 0375-9601/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 126, number I
PHYSICS LETTERS A
(AB)x and (A2B2)x have been calculated in the firstneighbor interaction approximation. For comparison we have calculated the (AB)x system as well in the second-neighbor interaction approximation. It should be mentioned that the number of atoms explicitly considered in the second-neighbor approximation of (AB)x is nearly equal to the number of atoms considered in the first-neighbor approximation of (A2B2)x.
3. Results and discussion
The resulting band structures are represented in fig. 1. There we find the bands which are situated in the valence and conduction band regions of the pure polymers. In the first column (from the left) are given the bands of (AB)x in the first neighbor approximation (N= 1). The second column contains the resuits for N = 2, which are drawn next to the results of (A:B2)x for N = 1 because of the argument at the end of section 2. (It can be seen that the N = 1, 2 resuits for (AB)x are almost identical, which demonstrates the convergence with the increase of N.) The last two columns contain our results for the valence
En(k) eV) +10
mmd --
d(A)
Tf"
d(A)
_ d(A)
}o
14 December 1987
and conduction bands of polyacetylene and polyethylene (compare with ref. [9]). The states which are mainly localized on the sublattice A (or B) are denoted in the figure by the letter A (or B), d refers to states which are delocalized over both sublattices, and d(A) assigns states which have components in both sublattices, but with considerably larger A-type components. The letter rt indicates the type of symmetry characteristic to the corresponding eigenvectors of the state considered. The width of the narrow (A) valence band of the system (AB)x is 0.0675 eV, while the widths of the narrow valence and conduction bands, (the d(A) bands), of the system (A2B2)x are 0.0564, 0.0606, 0.0375 and 0.0565 eV, respectively. Figs. 2a and 2b contain the details of the band structures of both systems. As can be qualitatively expected from negativefactor-counting calculations in the simple tight-binding approximation [ 10], there occurs in the copolymers a number of relatively narrow subbands, with gaps in between, in the region of the bands of the corresponding pure polymers. These subbands can be assigned to either delocalized (bulk) states which belong to both components of the copolymers, or to states which are localized only on one of the two (A or B) sublattices. It turns out that the highest valence band is of such a sublattice type, namely localized on the acetylene units, and has n symmetry. This is due to the fact that the A-type valence band is reaching to higher energies, than the B-type. Since these sublO, O
J
~
O
f
0,01
0,0'
n:
(AB)x (AB)x (A2B21x N:
1
-'°
__d(A) d(A)
2
1
Ax
10'0 1
E(eV)
E(eVl lO,0-
10,0-
20.0-
20.0-
-20
2
Fig. l. Band structure results for the (AB)x and (A:Bz)x copolymers where A=C2H2 and B=C2H4. Also the pure polymer results A,, B, have been included. The ordinate gives the band energies in eV. The types of the bands are indicated.
0,0
k
1Tlo
0,0
k
TTIo
Fig. 2. (a) Band structure of (AB)x system ( N = 1). (b) Band structure of (A2B2).,-system ( N = 1 ).
39
Volume 126, number 1
PHYSICS LETTERSA
b a n d s correspond to states which can be described by the A-type elementary cells with a translation vector which bridges the B-type cells, the widths of these subbands are very small. In the energy region where the A-type a n d B-type valence b a n d s overlap, wider s u b b a n d s of mixed eigenvector character appear. These s u b b a n d s are separated by a very small gap from each other in the case of (AB),, b u t they are overlapping in the case of (A2B2).,-. These results confirm the picture expected from simple model considerations for copolymers with partially overlapping bands. P s c h e n i c h n o v [ 6 ] has suggested that such copolymers could be used as molecular-electronic switches because the periodic superstructure permits resonant t u n n e l i n g which can be switched o f f b y a local change of the energy levels. O u r ab initio b a n d structure calculations have shown that in the case of copolymers with partially overlapping b a n d s really narrow s u b b a n d s are f o u n d which correspond to an array of potential well states and thus support the validity of the model considerations in a realistic example.
Acknowledgement The financial support by the "Deutsche Forschungsgemeinschaft" (Project-no. Se 463/1-1) a n d
40
14 December 1987
by the " F o n d s der Chemischen Industrie" is gratefully acknowledged.
References [ 1] K. Kanazawa, A.F. Diaz, M.T. Krounbi and G.B. Street, Synth. Metals 4 (1983) 119. [2] O. Ingan~is,B.O. Liedbergand W.U. Chang-ru, Synth. Metals 11 (1985) 239. [ 3] N. Holonyak Jr., R.M. Kolbas, R.D. Dupuis and P.D. Dapkus, IEEE J. Quantum Electron. 16 (1980) 170. [4] D.A.B. Miller, D.S. Chemla, T.C. Damen, A.C. Gossard, W. Wiegmann, T.H. Wood and C.A. Burrus, Appl. Phys. Lett. 45 (1984) 13. [ 5 ] D.S. Chemla, D.A.B. Miller, P.W. Smith, A.C. Gossard and W.Wiegmann,IEEE J. Quantum Electron. 20 (t984) 265. [6] E.A. Pschenichnov, Sov. Phys. Solid State 4 (1962) 819; J.R. Barker, in: Proc. 2nd Int. Workshopon Molecular electronic devices, ed. F.L. Carter (Dekker, New York, 1983). [7] G. Del Re, J. Ladik and G. Bicz6, Phys. Rev. 155 (1967) 977; J.-M. Andr6, L. Gouverneur and G. Leroy, Int. J. Quantum Chem. 1 (1967) 427, 451. [8 ] W.J. Hehre, R.F. Stewart and J.A. Pople, J. Chem. Phys. 51 (1969) 2657. [9] J. Ladik, J.-M. Andr6 and M. Seel, Quantum chemistry of polymers - solid state aspects ( Reidel, Dordrecht, 1984). [ 10] M. Seel, Chem. Phys. 43 (1979) 103.