Optical Properties of Nanostructures

Copyright © 2011 American Scientific Publishers All rights reserved Printed in the United States of America

Journal of Computational and Theoretical Nanoscience Vol. 8, 1–6, 2011

Optical Properties of Nanostructures Vjekoslav Sajfert1 ∗ , Stevo Ja´cimovski2 , Jovan P. Šetrajˇci´c3 , Ljiljana Maškovi´c2 , Nikola Bednar4 , Nicolina Pop5 , and Bratislav Toši´c6 1

University of Novi Sad, Technical Faculty “M. Pupin”, 23000 Zrenjanin, Djure Djakovica bb, Serbia 2 Academy of Criminalistic and Police Studies, Belgrade, Zemun, Cara Dusana 196, Serbia 3 University of Novi Sad, Faculty of Sciences, 21000 Novi Sad, Trg Dositeja Obradovica 3, Serbia 4 University of Novi Sad, Faculty of Technical Sciences, 21000 Novi Sad, Trg Dositeja Obradovica 6, Serbia 5 “Politehnica” University of Timisoara, Piata Victoriei Number:2, Post Code: 30006 6 Vojvodina Academy of Science and Arts, Novi Sad, Dunavska 37, Serbia We analyzed linear molecular chain with exciton excitations for the case when number of excitons is not conserved. The results obtained are in some sense amazing. The dispersion law of finite chain is surface depending on two independent angles. The same conclusion is valid for concentrations of exitons and exciton pairs. As it was expectable physical characteristics of the finite chain depend on spatial coordinates. All results are compared to corresponding results of infinite chain.

Keywords: Non-Conservation of Excitons, Finite Chain, Surface Like Dispersion Law.

Optical excitations in condensed matter are called excitons. There are two types of excitons. One of them appear in semiconductors and they are called Mott-Wannier excitons.1 The excitons appearing in molcular crystals are called Frenkel excitons.2 3 Their main characteristics is that both are bounded particle–hole pairs. In semiconductors hole is in valence zone while particle (electron) is in conductive zone. The connection between elements of this pair remains stable. As excitation moves as electroneutral complex. The electron–hole distance is relatively high. It is the reason for Vanier Mott excitons1 to be called “high” radius excitons. In molecular crystals (anthracene, naphtacene, naphtaline, etc.) electron and hole remain at the molecule. The electron wave functions of neighbour molecules have weak covering. Since mentioned pairs remain at molecule these excitons are called “low radius excitons.” Our investigations will be concerned to Frenkel excitons, mainly. Theory of Frenkel excitons was developed in works of Frenkel, Davydov, Agranovich and Knox.2–6 There are two types of Frenkel excitons. One type is caused by eletron subsystem excitations of an isolated molecules, while second is caused by excitations of internal molecular oscillations. The last are called vibrons and their Hamiltonian is ∗

Author to whom correspondence should be addressed.

J. Comput. Theor. Nanosci. 2011, Vol. 8, No. 11

expressed in Boze operators. The excitons which arise due to excitations of electron subsystem have mixed statistics of operators which create and annihilate excitons. At one lattice point they have Fermi statistics, while for different lattice points they are more close to bozons. Such problem appeared in theory of magnetism much earlier than in theory of excitons, but after appearence of lasers the question about kinematical and dynamical interactions of excitons became actual. For two-level molecular excitations, exciton operators are Pauli operators. For multilevel scheme of molecular excitations exciton operators have more complicated kinematics than paulion’s ones.7 The part of optics whose subjects are exciton interactions is called nonlinear optics.8 9 The problems of nonlinear optics are appearance of higher harmonics, multiphoton absorbtion,10 Uhrbach’s role6 12 and problem of non-conservation of number of excitons.5 13 14 It should be noted that interactions between molecules in molecular crystals are mainly of dipol–dipol type. It causes non-analicity of dispersion laws (energy of excitons for k = 0 depends on direction of propagation).5 All mentioned problems become more complicated in nanostructures and in strucures with random disturbed symmetry. We shall start our investigations with the problem of short one-dimensional macromolecul whose number of excitons is not conseved. Such macromolecules are, by Sent George, very important for biophysical processes.5

1546-1955/2011/8/001/006

doi:10.1166/jctn.2011.1957

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1. INTRODUCTION

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Optical Properties of Nanostructures

  + BN −1 BN + BN −2

2. NON-CONSERVATION OF EXCITONS IN SHORT LINEAR CHAIN We shall consider linear chain containing N molecules where N ∈ 3     30 molecules. In this analysis will be used ASQ (Approximate Second Quantisation) method.15 It means that we analyse only quadratic part of exciton Hamiltonian, expressed in Bose operators B + and B. The non-conservation of excitons will  be taken into account,  i.e., the fact that H  n Bn+ Bn = 0. Consequently, the Hamiltonian of the chain, taken in the nearest neighbours approximatin, is given by H=

N 

Bn+ Bn +

n=0

 Xnn+1 + Xnn−1 Bn+ Bn

 +

Bn Yn+1 Bn+1 + Yn−1 Bn−1



n=0 N   1 + + Bn+ Znn+1 Bn+1 + Zn−1 Bn−1 2 n=0   + Bn Znn+1 Bn+1 + Znn−1 Bn−1

RESEARCH ARTICLE

(1)

In this Hamiltonian  ∼ 3−5 eV is excitation of an isolated molecul while X Y and Z are matrix elements of dipole–dipole interactions and they are about 100 times less than . Since number of molecules N is small the boundary conditions must be taken into account. Those boundary conditions are given by

(8)

the system of six partial difference equations 1 ≤ n ≤ N −1 EGnm   =

i  +  + 2XGnm   2 nm + Y Gn+1m   + Gn−1m   + ZDn+1m   + Dn−1m  

(10)

EDnm   = − + 2XDnm   − Y Dn+1m   + Dn−1m   − ZGn+1m   + Gn−1m  

(11)

i  +  + XG0m   2 0m + YG1m   + ZD1m  

(12)

n=0

F ∈ X Y  Z

H = H 1 + H2

EG0m   =

(3)

ED0m   = − + XD0m  

(4)

− YD1m   − ZG1m  

(13)

i  +  + XGN m   2 N m + YGN −1m   + ZDN −1m  

(14)

n=N

where N 

B0+ B0 + XB0+ B0 + Y B0+ B1 + BN+ BN

EGN m   =

n=0

+ XBN+ BN + Y BN+ BN +1 + 

N 

Bn+ Bn

EDN m   = − + XDN m  

n=0

+ 2X

N  n=0

Bn+ Bn + Y

N 

  Bn+ Bn+1 + Bn−1

(5)

n=0

    1  H2 = Z B0+ B1 + B1+ B2+ + B0+ + B0 B1 + B1 B2 + B0 2   1  + Z BN+ BN −1 + BN+−1 BN+ + BN+−2 + BN BN −1 2 2

(7)

tBn+ t Bm+ 0

Dnm t =

and taking into account the boundary conditions (2) we write Hamiltonian (1) in the form

H1 =

Gnm t = tBn t Bm+ 0

(2)

where F stands for X Y and Z. Introducing notation Fnn±1 = F 

The system (4) will be investigated by means of Green’s functions16−18

−

+

F0−1 = FN N +1 = 0

(6)

where t is Heaviside step function. Differentiating (7) and (8) with respect to t and using equations of motion for operators B and B + , we obtain after Fourier tranformation19  +

F t = dte−i t F   F ∈ G D (9)

n=0

N 

+

N  

N      1  + + Z B + B + + Bn−1 + Bn Bn+1 + Bn−1 2 n=0 n n+1

− YDN −1m   − ZGN −1m  

(15)

Our goal is reducing of the system of Eqs. (10)–(15) to the system containing only two equations which are valid for all values of n, i.e., for n = 0 1 2     N . This is practically impossible with this system and therefore we are going over to a symmetrised system, containing sums and J. Comput. Theor. Nanosci. 8, 1–6, 2011

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Optical Properties of Nanostructures

differences of functions G and D. This symetrised system is the following: 1 ≤ n ≤ N −1  Y + Z Sn+1m   + Sn−1m  

sinn + 2 +

X + Y −Z

2

2X sinn + 1 Y −Z sin n = 0

(30)

the system of equations reduces to i + Snm   − ERnm   = −  2 nm   Y − Z Rn+1m   + Rn−1m   + Rnm   − ESnm   = −

(16)

N +1 

=− −

N +1 

i  2 0m (18)

Y − ZR1m   +  − xR0m   − ES0m   = −

i  2 0m (19)

i  2 N m

=1

i   2 nm

(21)

(32)

+ = 2Y + Z cos  +  + 2X

(33)

−

(34)

= 2Y − Z cos  +  + 2X

 m  =  m  

(35)

 n  =  m  

(36)

For Kronecker symbol will be taken the representation N +1  

 L n + M n  m

n m ∈ 0 1     N 

=1

(22)

R = G−D

(23)

 =  + 2X

(24)

and It can be easily concluded that with the transformations  (25) Snm   =  m L n

(37) The values of functions  m can be determined from the system of Eq. (37). The set of these functions are given on the Table I and b. Substituting (35)–(37) we obtain the system of equations for unknown functions  and : +

L n    L n + M n 



(26)

−E



where X sin n L n = sinn + 1 + Y +Z X M n = sinn + 1 + sin n Y +Z

n m ∈ 0 1     N 

where

(20)

S = G+D

 m M n

− M n +  m 

=1

=−

nm =

Rnm   =

N +1 

(31)

In further we shall take

where



n m ∈ 0 1     N 

(27) (28)

−E

M n i    = − L n + M n 2

(38)

L n    L n + M n  M n i    = − L n + M n  2

(39)

   =

i L n + M n E + − 2

L n E 2 − E2

(40)

   =

i L n + M n E + + 2

M n E 2 − E2

(41)

+− The solutions are:

and 2X sinn + 2 + sinn + 1 Y +Z 2

X + sin n = 0 Y +Z J. Comput. Theor. Nanosci. 8, 1–6, 2011

(29)

3

RESEARCH ARTICLE

−ESN m   = −

i   2 nm

EL n m  +

n=N Y + ZSN −1m   +  − xSN m  

EM n m 

=1

(17)

n=0

i −ERN m   = −  2 N m Y − ZRN −1m   +  − xRN m  

N +1 

=1

i  2 nm

Y + ZS1m   +  − xS0m   − ER0m   = −

+ L n +  m  −

Sajfert et al.

Optical Properties of Nanostructures Table 1a.

Values of  m for X = 50 Y = 400 Z = 300. n



1

2

3

4

5

6

7

8

9

10

11

1 2 3 4 5 6 7 8 9 10 11

0.0781 0.1498 0.2093 0.2518 0.2743 0.2757 0.2570 0.2212 0.1731 0.1176 0.0592

−0.0982 −0.2201 −0.3772 −0.5616 −0.7455 −0.8893 −0.9545 −0.9178 −0.7789 −0.5591 −0.2901

0.490 0.923 1.291 1.639 2.015 2.406 2.712 2.785 2.514 1.891 1.010

−1.194 −2.477 −3.801 −5.020 −6.061 −6.957 −7.678 −7.954 −7.378 −5.718 −3.117

3706 7264 10778 14326 17647 20347 22185 22770 21219 16672 9205

−10.533 −21.193 −31.645 −41.662 −51.157 −59.342 −64.839 −66.241 −61.589 −48.608 −27.018

31859 63427 94815 12527 15349 17783 19471 19892 1846 1458 81292

−102.07 −203.93 −304.39 −402.05 −493.12 −570.92 −625.07 −638.98 −592.67 −467.98 −261.26

37624 75101 11217 14812 18166 21037 23026 23543 21836 17239 96276

23707 47329 70679 93339 114470 132560 145100 148350 137610 108620 60672

83648 167000 249390 329340 403910 467750 511980 523450 485550 383270 214080

8

9

10

11

2430 4869 7316 9749 12099 14223 15843 16495 15537 12375 6930

−3270 −6552 −9845 −13117 −16280 −19138 −21318 −22195 −20906 −16651 −9324

−25137 −50365 −75676 −100830 −125150 −147110 −163870 −170610 −160700 −128000 −71675

272390 545760 820040 1092600 1356100 1594100 1775700 1848800 1741400 1387000 776680

Table 1b.

Values of  m for X = 50 Y = −400 Z = 600.

RESEARCH ARTICLE

n 

1

2

3

4

5

1 2 3 4 5 6 7 8 9 10 11

0.062 0.118 0.164 0.196 0.211 0.207 0.184 0.148 0.104 0.062 0.028

00351 00489 00289 −00223 −00858 −01323 −01340 −00820 00001 00612 00588

−0.543 −1.138 −1.792 −2.457 −3.054 −3.522 −3.837 −3.972 −3.810 −3.135 −1.807

562 1117 1669 2222 2767 3261 3627 3765 3545 2834 1596

−357 −717 −1079 −1436 −1782 −2095 −2335 −2430 −2288 −1823 −1022

where E =



6 1864 3733 5609 7474 9276 10903 12146 12647 11911 9487 5314

+ − =  + 2X2 + 4Y 2 − Z 2  cos  cos 

7 −785 −1572 −2362 −3147 −3906 −4592 −5115 −5326 −5016 −3995 −2237

Dnm   =

+ 2 + 2X Y + Z cos  + Y − Z cos  1/2

(42)

is the energy of the elementary excitations in the chain. In order to find the functions Gnm   and Dnm   we include (40) and (41) in (35) and (36) and the obtained results substitute into (25) and (26). So we find  i   E + − Snm   = L n + M n  m 2 (43) 2  E − E2

i  L n + M n  m 2  4

− + + −  /  −  /  × −  − /+  − + /−  (46) − + 

where

E (47)  The spectral intensities of the function G and D are given, respectively by:  =

Gnm  + i − Gnm  − i

 i   E + + L n + M n  m 2 (44) 2  E − E2

IG =

Since G = 1/2S + R and D = 1/2S − R we obtain, after substitution (43) and (44), that

ID =

Rnm   =

Gnm   =

4

i  L n+M n  m 2  4

2+ − /+ + + /−  × −  2− − /+ + + /−  (45) + + 

 /k T 

e→0+B − 1 Dnm  + i − Dnm  − i  /k T 

e→0+B − 1

(48) (49)

Using the formula   1 1 ∓ i  ± 0 = ± 0 ± i ± 0

(50)

and substituting (45) into (48) and (46) into (49), we find:  L n+M n  m IG = 4  J. Comput. Theor. Nanosci. 8, 1–6, 2011

Sajfert et al.

Optical Properties of Nanostructures

2+ − /+ + + /−  ×  − 0  e /kB T  −1  2− − /+ + + /−   +  + (51) 0 e− /kB T −1

and concentration of pairs Bn+ Bn+ . So we obtain Bn+ Bn =

and ID =

 L n+M n 



×

4

N +1 

L n + M n  n 4 =1     − + E coth − 4 (53) × + + − 2kB T

and

 m

 − /+ − + /−    − 0  /k T  B e −1 − +   / − + /−   +  + 0 (52) e− /kB T  −1

Bn+ Bn+ =

N +1 

L n + M n  n 4 =1     E − + × coth − (54) + − 2kB T

The correlation functions can be determined by multiplaying (50) and (51) with ei t and integration with respect to from − to + . The correlation functions, taken for m = n and t = 0 give the concentration of excitons Bn+ Bn

The results obtained for E  Bn+ Bn and Bn+ Bn+ are very interesting since the energy of excitations as well as concentrations of excitons and the pairs are the functions of two arguments,  and , and in coordinte system   and E (or Bn+ Bn  Bn+ Bn+ and E) represent surfaces. This

(a)

(a)

6500

6600

6400

6600

7000 6400

6300

6500

6400

6200

6200 6100 6000

5500 5800 5000 5600

5900

4500 3

5800

5600 0

6000

E

6000

5800

5400

5700

2

χ(rad)

1

2 3

1 0

4 (b) 2.1 × 10

(b)

2.1

2.08

2.08

2.06

2.06

2.04

2.04

2.02

5200

2

ϕ(rad) 1

ϕ(rad)

0

3

2

3 1

Eγ

6000

χ(rad)

0

× 104

2.02

Eγ 2

2

1.98

1.98

1.96

1.96

1.94

1.94

1.92

0

0.5

1

1.5

2

2.5

3

γ(rad)

Fig. 1. (a) Dispersion law of finite chain with negative effective mass for set of parameters X = 50 Y = 400 Z = 300. (b) Dispersion law for infinite chain for set of parameters X = 50 Y = 400 Z = 300.

J. Comput. Theor. Nanosci. 8, 1–6, 2011

1.92

0

0.5

1

1.5

2

2.5

3

γ(rad)

Fig. 2. (a) Dispersion law of finite chain with positive effective mass for set of parameters X = 50 Y = −400 Z = 600. (b) Dispersion law of infinite chain for set of parameters X = 50 Y = −400 Z = 600.

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6200

E

6800

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Optical Properties of Nanostructures

essentially differ from behaviour of the same characteristics for infinite chain where E, B + B and B + B + are represented as lines in function of the angle  = ak. In infinite chain energy is given by E =  + 2X + 2Y cos 2 − 4Z 2 cos2  (55) In order to compare to dispersion laws of finite and infinite chain we are including dispersion law diagrams if Figures 1(a, b) and 2(a, b). The concentrations of infinite chains are

1   + 2X + 2Y cos  Bn+ Bn  = N  2E   + 2X + 2Y cos  − 1 (56) × coth 2kB T and Bn+ Bn+ 

 1  Z cos  Z cos  =− coth + 1 (57) N  E kB T

RESEARCH ARTICLE

The second specific characteristics of the finite chain is dependence of consentrations on confugurational index n, but it was expected since in broken symmetry systems physical characteristics are dependent on spatial configuration.

3. CONCLUSION The analysis of non-conserving excitons in short onedimensional chain has shown that it fundamentally differs from the corresponding infinite linear chain. The first difference is the fact that physical characteristics of finite chain are dependent on spatial indices. This result was expectable since the symmetry of finite chain is broken. The second difference is more amazing: physical characteristics of finite chain depend on two independent parameters and set of energies as well as set of concentration points forms discrete surface-like diagram of points. In the same time the mentioned characteristics of infinite chain are curves depending on one angle  = ak. Physical

explanation of this fact is not a simple problem. In our opinion it comes due to the presence of usual excitons and exciton pairs which are behaving in some sense independently. The diagrams of dispersion laws for finite as well as infinite chain are given on Figures 1(a, b) and 2(a, b) for different values of exciton parameters X Y and Z. On these figs the differences between finite an infinite chain are more noticeable. Finally it will be pointed out that problem of autoreduction is avoided in given analysis since it was taken that X/Y + Z and X/Y − Z are taken less than unit. The effect of autoreduction will be the subject of further analyses of excitons.

References 1. C. Kittel, Introduction to Solid State Physics, Wiley, New York (2005). 2. J. Frenkel, Phys. Rev. 37, 17 (1931). 3. J. Frenkel, Phys. Rev. 37, 1276 (1931). 4. A. S. Davydov, Theory of Molecular Excitons, Nauka, Moscow (1978). (In Russian). 5. V. M. Agranovich, Theory of Excitons, Nauka, Moscow (1978). (In Russian). 6. R. Knox, Theory of Excitons, Academic Press, New York (1963). 7. D. I. Lalovi´c, B. S. Toši´c, and R. B. Žakula, Phys. Rev. 178, 1472 (1969). 8. D. L. Mills, Nonlinear Optics: Basic Concepts, Springer-Verlag, Berlin Heidelberg (1991). 9. S. A. Ahmanov and R. V. Hohlov, UFN 88, 439 (1966). 10. B. S. Toši´c, FTT 9, 1773 (1967). 11. U. F. Kozmidis-Luburich and B. S. Toši´c, Optical Excitations in Material Mediums, University of Novi Sad, Novi Sad (2000). 12. F. Urbach, Phys. Rev. 92, 1324 (1953). 13. N. Blombergen, Nonlinear Optics, Mir, Moscow (1966). (In Russian) 14. R. Loudon, Quantum Theory of Light, Oxford University Press, Oxford (1983). 15. N. N. Bogolyubov, Selected Works: Quantum and Classical Statistical Mechanics, Gordon and Breach (1990). 16. V. Sajfert, J. Šetrajˇci´c, D. Popov, and B. Toši´c, Physica A 353C, 217 (2005). 17. V. Sajfert, J. Šetrajˇci´c, B. Toši´c, R. --D aji´c, and Czech, J. Phys. 54, 975 (2004). 18. V. Sajfert, S. Ja´cimovski, and B. Toši´c, Journ. of Luminescence 128, 1459 (2008). 19. R. N. Bracewell, The Fourier Transform and its Applications, Mc Graw Hill, Boston, Burr Ridge, New York, San Francisco, Madrid (2000).

Received: 18 January 2011. Accepted: 11 February 2011.

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