soliton excitations of a one-dimensional molecular-crystal model with the dispersion term

phys. stat. sol. (b) 208, 435 (1998)

Subject classification: 71.38.�i; 63.20.Kr

Soliton Excitations of a One-DimensionalMolecular-Crystal Model with the Dispersion Term

Hao Chen (a) and Yuan Chen (b)

(a) Department of Physics, South China Normal University,Guangzhou 510631, People's Republic of China

(b) Department of Physics, Zhongshan University,Guangzhou 510275, People's Republic of China

(Received July 22, 1997; in revised form May 18, 1998)

An improved solution for soliton excitations of a one-dimensional molecular-crystal model withthe dispersion term is found. Under the approximation of neglecting the dispersion term, the im-proved solution tends to the usual polaron solution. We calculate the lattice, displacement of theself-trapped state, the self-trapped well of the electron, the density of current carried by solitonexcitations and the width, the peak the effective mass and the binding energy of the soliton excita-tion in the model. The differences between the soliton excitation in the model and the usual polaronare pointed out.

1. Introduction

As is now known, an electron (or hole) in a molecular-crystal medium may lower itsenergy by locally distorting the crystal lattice surrounding it [1]. Due to the electron±phonon interaction, such a lattice deformation can produce a potential well in whichthe electron is trapped. This entity, the ªself-trappedº electron together with its inducedlattice distortion, is commonly called the polaron. This has attracted a lot of researchinterest until the present time [2].

The possibility of self-trapping in a one-dimensional electron±phonon system wasfirst studied by Holstein [3]. Further research on polarons was done by many authors[4]. In this paper, our purpose is to make an investigation on a one-dimensional mole-cular-crystal model (MCM) with the dispersion term and find an improved solution forthe solution excitation in the model. We calculate the lattice displacement of the self-trapped state, the self-trapped well of the electron, the density of current, and thewidth, the peak, the effective mass and the binding energy of the soliton excitation(BESE) in the model. This paper is organized as follows. In Section 2, the equation ofmotion for soliton excitations in MCM and its solutions are given. The various para-meters of soliton excitations are calculated and discussed in Section 3, and our conclu-sions in Section 4.

2. The Equation of Motion and Solutions

The Hamiltonian of the MCM with the dispersion term is given by [5]

H � He �Hint �Hp ; �1a�

Hao Chen and Yuan Chen: Soliton Excitations of a 1D Molecular-Crystal Model 435

He �Hint � ÿ �h2

2m

@2

@y2� PN

n�1V�yÿ Rn; un� ; �1b�

Hp �PNn�1

ÿ �h2

2M

@2

@u2n

� M

2w2

0u2n �

M

2w2

1unun�1

!�1c�

which describes the electrons interacting with lattice optical vibrations through the de-formation potential in one dimension, where y and m are, the electron coordinate andmass, respectively, a; Rn � na, and un denote, lattice spacing, the position and displace-ment of a molecule at the n-th site, respectively, V�yÿ Rn; un� is the interaction poten-tial energy between an electron and a molecule, M is the reduced mass of a molecule,N is the number of molecules in the chain, w0 is the Einstein frequency, and the w2

1-proportional term is the so-called dispersion term. Following Holstein [5], we assumethe wavefunction y of H is

w �Pn

bnjn�y; un� : �2�

Here bn, which depends on u1; . . . ; uN and time t, is the probability amplitude of theelectron at the n-th site, and j�y; un�, which describes the local orbital of the molecule,satisfies the eigenequation

ÿ �h2

2m

@2

@y2� V�yÿ Rn; un�

" #jn�yÿ Rn; un� � E�un� jn�yÿ Rn; un� ; �3�

where the eigenvalue E�un� only depends on the displacement of the molecule atthe n-th site. Making the linear approximation, we have

E�un� � ÿAun : �4�From the SchroÈ dinger equation

i�h@

@tw � Hw ;

by a method similar to the tight-binding theory, the equation of motion for bn can beobtained as follows:

i�h@bn

@t� �Hp ÿAun� bn � Jn;n�1bn; n�1 � Jn; nÿ1bn; nÿ1 ; �5�

where

Jn;m ��

j*n�yÿ Rn; un� V�yÿ Rn; un� jm�yÿ Rm; um� dy �6�

is the exchange integral, and the term ÿAun plays the role of an interaction potentialbetween the electrons and the lattice. Omitting the dependence of Jn; n�1 on um, lettingJn; n�1 � ÿJ, and taking the adiabatic approximation, i.e., dropping the kinetic energyterm of the lattice vibration in Hp, Eq. (5) becomes

i�h@bn

@t� Mw2

0

2

Pm

u2m �

Mw21

2

Pm

umum�1 ÿAun

� �bn ÿ J�bn�1 � bnÿ1� : �7�

436 Hao Chen and Yuan Chen

Assuming bn is a smooth function of its lattice-site ªargumentº n, and applying thecontinuum limit

bn�t� � b�x; t� ; bn�1 � b�x; t� � @b

@xa� 1

2@2b

@t2a2 � . . . ;

un�t� � u�x; t� ;Eq. (7) changes into

i�h@b

@t� Mw2

0

2a

�1ÿ1

u2�z� dz� Mw21

2a

�1ÿ1

u�z� u�z� a� dzÿAu

24 35 b

ÿ2Jbÿ Ja2 @2b

@x2: �8�

We now look for the solution of Eq. (8) in the form

b�x; t� � f�h� ei �kxÿ�E=�h� t� ; �9�where f is a real function of h and h � xÿ vt. Equation (8) turns into

k � �hv

2Ja2�10�

and

Ja2 d2f

dh2

ÿ Mw20

2a

�1ÿ1


2a

�1ÿ1

u�z�u�z� a� dzÿAuÿ 2J � Jk2a2 ÿ E

24 35f � 0 :

�11�The normalization condition and the minimal energy condition give rise to the minimalenergy displacement u0�x�, corresponding to the minimal energy E�u0�x��, to be

u0�x� � Aa

Mw20

f20 ÿ

w21

2w20

�u0�x� a� � u0�xÿ a�� : �12�

It is interesting only for the narrow-band case, i.e. w1 � w0. In this situation, an itera-tive solution for Eq. (12) is given by

u0�h� � Aa

Mw20

1ÿ w21

w20

� �f2

0�h� ÿAa3w2

1

Mw40

df0

dh

� �2

ÿ Aa3w21

Mw40

f0d2f

dh2�O

w1

w0

� �3" #

:

�13�Combining Eqs. (11) and (13), we obtain the equation for f0

�1ÿ gf20�

d2f0

dh2ÿ gf0

df0

dh

� �2

� 2mf30 ÿ lf0 � 0 ; �14�

where

g � A2aw21

JMw40

; m � A2

2JMw20a

1ÿ w21

w20

� �; l � e

Ja2;

Soliton Excitations of a One-Dimensional Molecular-Crystal Model 437

and

e � Mw20

2a

�1ÿ1


1

2a

�1ÿ1

u�z� u0�z� a� dzÿ 2J � Jk2a2 ÿ E : �15�

Equation (14) is a nonlinear equation and has an exact solution, i.e., the soliton solu-tion, as

1��lp arcosh

2lfÿ20 ÿ �lg� m�jlgÿ mj ÿ

��g

m

rarcosh

2mgf20 ÿ �lg� m�lgÿ m

�� 2�hÿ h0�

�16�with h0 � x0 ÿ vt0 denoting the position of the center of the soliton excitation being x0

at time t0. As w1 � w0, one has g� 1 and g < m=l. Then the second term on the left-hand side of Eq. (16) can be neglected and an approximate soliton solution of Eq. (14)is

f0�h� �m

lÿ g

� �cosh2

��lp�hÿ h0� � g

h iÿ1=2: �17�

In the following, one will see that Eqs. (16) and (17) are very different from the usualpolaron solution in MCM in which the w2

1-proportional term is absent [3].Our following results are based on Eq. (17). The normalization condition for f0

yields

l � m

gtanh2

��mgp

2: �18�

Thereby f0�h� can be rewritten as

f0�h� � g cosech2��mgp

2cosh2 hÿ h0

Ls� g

� �ÿ1=2

; �19�

where

Ls � L0s 1ÿ w2

1

w20

� �ÿ1

q�a� �20�

is the width of the soliton excitation,

L0s � 4JMw2

0a=A2 �21�is the width of the usual polaron [3], and hereafter

q�a� � a coth a ; a � ��mgp

=2 : �22�

3. Results and Discussion

The peak of the soliton excitation is

Ps � P0s 1ÿ w2

1

w20

� �1=21

q�a� ; �23�


where

P0s �

A2

8JMw20

� �1=2

�24�

is the peak of the usual polaron [3]. Substituting Eq. (19) into Eq. (13) and correctingto the first order in a and the second order in w1=w0, one can get the lattice displace-ment of the self-trapped state

u0�h� � Aa

Mw20

1ÿ w21

w20

� �g cosech2

��mgp

2cosh2 hÿ h0

Ls� g

� �ÿ1

; �25�

but the self-trapped potential well of the electron is given by

V�u0� � ÿAu0�h� � ÿ A2a

Mw20

1ÿ w21

w20

� �g cosech2

��mgp

2cosh2 hÿ h0

Ls� g

� �ÿ1

:

�26�From Eq. (8), the density of current carried by soliton excitations is got as

j�h� � ÿevf20�h� � ÿev g cosech2

��mgp

2cosh2 hÿ h0

Ls� g

� �ÿ1

; �27�

where ÿe is the charge of an electron. It is evident from Eqs. (19) and (25) to (27) thatthe lattice displacement of the self-trapped state and the self-trapped potential well ofthe electron and the density of current move together with the soliton excitation atsame velocity v. The maximum of the lattice displacement of the self-trapped state andthe depth of the self-trapped potential well of the electron are

umax � u0max 1ÿ w2

1

w20

� �21

q2�a� �28�

and

Vd � V0d 1ÿ w2

1

w20

� �21

q2�a� ; �29�

respectively, where

u0max � 2Ja2=AL02

s and V0d � 2Ja2=L02

s �30�are, the same as in the usual MCM [3]. The maximum of the density of current is

jmax � j0max 1ÿ w2

1

w20

� �1

q2�a� ; �31�

where

j0max � ÿ

evA2

8JMw20

�32�

is that as in the usual MCM.As q�0� � 1, it is easy to verify that in the limit w1 � 0, i.e., g � 0

Ls � L0s ; Ps � P0

s ; umax � u0max ; Vd � V0

d : �33�


and f0�h�, u0�h� and V�u0� become

f0�h� � P0s sech

hÿ h0

L0s

; �34�

u0�h� � u0max sech2 hÿ h0

L0s

; �35�

and

V�u0� � ÿV0d sech2 hÿ h0

L0s

�36�

which coincide with the results as in the usual MCM [3]. On the other hand, asq�a� > 1 for a 6� 0, we may gain at once from Eqs. (20), (23), (28), (29) and (31), whenw1 6� 0,

Ls > L0s ; Ps < P0

s ; umax < u0max ; Vd < V0

d ; and jmax < j0max :

�37�That is to say, the existence of the w2

1-proportional term contained in the Hamiltoniancauses the width and the peak of the soliton excitation, to widen and to lower, respectively,the maximum of the lattice displacement of the self-trapped state to reduce, the depth ofthe self-trapped potential well of the electron to shorten, and the density of current carriedby soliton excitations to reduce, which is unfavorable to stability of the soliton excitation.

From Eqs. (10) and (15), the energy of the soliton excitation is

Es�v� � Es�0� � 12 msv

2 ; �38�where

ms � �h2=2Ja2 �39�is the effective mass of the soliton excitation, and

Es�0� � ÿEp � E0 �40�is the rest energy of the soliton excitation,

E0 � ÿ2J �41�is the energy of the electron in the bottom of Bloch band, and

Ep � eÿMw20

2a

�1ÿ1

u20�z� dzÿ Mw2

1

2a

�1ÿ1

u0�z� u0�z� a� dz �42�

is the BESE. From Eqs. (15), (18) and (25), we can obtain

Ep � Ja2m

gtanh2

��mgp

2ÿ A2a

2Mw20

1ÿ w21

w20

� �2

� 12g

1� tanh2��mgp

2

� �ÿ 1

m1=2g3=2tanh

��mgp

2

� �

ÿ Jg

21ÿ w2

1

w20

� �2f ÿ coth

��lp

f 2 ÿ g2 cosh2��lp ; �43�


where

f � 12

m

l� g

� �; g � 1

2m

lÿ g

� �: �44�

The first term on the right-hand side of Eq. (43) is the potential energy of the electronin the potential well, and the opposite sign of sum of the second and third terms is thedeformation potential energy of the lattice. Expanding the first two terms on the right-hand side of Eq. (43) to the fourth order in w1=w0, we find the BESE to be

Es � E0s ÿ

A8w21

48J3M4w100

ÿ A4w41

12JM2w80

� A8w41

320J3M4w120

ÿ E0 ; �45�

where

E0p �

A4

48JM2w40

�46�

is the BESE in the usual MCM [3], and

E0 � Jg

21ÿ w2

1

w20

� �2f ÿ coth

��lp

f 2 ÿ g2 cosh2��lp > 0 : �47�

As w1 � w0,

Ep < E0p : �48�

That is to say, the w21-proportional term contained in the Hamiltonian makes the BESE

lower, which is also harmful to the stability of the soliton excitation.

4. Conclusion

In summary, we study the soliton excitation in the MCM with the w21-proportional term

which affects the self-trapped well of the electron and properties of soliton excitations,e.g. the peak, the width and the binding energy, etc., and plays a disadvantageous roleto the stability of soliton excitations. Our results are useful to understanding polarons.Further applications of our results will be given in a forthcoming paper.

Acknowledement This work is supported in part by the Natural Science Foundation ofGuangdong Province of China.

References

[1] L. D. Landau, Z. Phys. Sovietunion 3, 664 (1933).[2] H. B. Schuttler and T. Holstein, Ann. Phys. 166, 93 (1986).

T. Holstein and L. A.Turkevich, Phys. Rev. B 38, 1901, 1923 (1988)T. Holstein, Mol. Cryst. Liq. Cryst. 77, 235 (1981).A. H. Castro Neto and A. O. Caldeira, Phys. Rev. B 6, 8858 (1992).

[3] T. Holstein, Ann. Phys. 8, 325 (1959).[4] G. Whitfield and P. B. Show, Phys. Rev. B 14, 3346 (1976).

P. B. Show and G. Whitfield, Phys. Rev. B 17, 1495 (1978).N. Tokuda, Solid State Commun. 35, 1025 (1980); J. Phys. C 13, L73 (1980).D. K. Campbell, A. R. Bishop, and K. Fesser, Phys. Rev. B 26, 6862 (1982).

[5] T. Holstein, Ann. Phys. 8, 343 (1959).L. Friedman and T. Holstein, Ann. Phys. 21, 494 (1963).


soliton excitations of a one-dimensional molecular-crystal model with the dispersion term

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