Download A Theory of Frenkel Exeitons Using a Two-Level

1848
Progress of Theoretical.Ph ysics, Vol. 50, No. 6, December 1973
A Theory of Frenkel Exeitons Using
a Two-Leve l-Atom Model
Shozo T AKENO and Mahito MABUCHI
Departmen t of Nuclear Engineering , Kyoto University, Kyoto
(Received June 15; 1973)
A formal, rigorous theory of Frenkel excitons interacting with an external electromagneti c
field is doveloped within the framework of a two-level•atom model in which the dipole
approximation is employed in describing the atom-field (taken as a c-number) interaction.
The model exciton Hamiltonian is shown to be identical in form. with the Heisenberg model
Hamiltonian for S=l/2 spins with anisotropic long-range exchange interactions. Instead of
treating excitons as a non-ideal Bose gas, atomic polarization operators and level population
difference are taken up as two fundamental quantities. to describe Frenkel excitons. A pair
of equations satisfied by these two quantities are derived. It is shown that Frenkel excitons
are generally expressed as quantum-mech anical nonlinear polarization waves, the nonlinearity
being characterized by the level population. A discussion is given · on the interrelationsh ip
between such polarization waves and those derivable from the classical Drude-Lorentz model.
\Vith the aid of several approximation procedures similar to those employed in nonlinear
optics, a set of equations describing nonlinear exciton-photon coupled modes are derived.
Brief discussion is .given on the eigenfrequenci es. of nonlinear Frenkel excitons and nonlinear
polaritons. The .existence of nonlinear exciton-photon coupled modes whose. properties are
different from those of polariton modes is suggested.
§ 1.
Introductio n
A number of theoretical studies have been made on the properties of Frenkel
excitons in molecular crystals 1l and in certain polymers 2l and macromolec ules 8l as
well. Employed here, implicitly but extensively, are·"interac ting"-two-le vel-atom
(or molecule) models, in which only two of energy levels of a given atom, corresponding to the ground state and an excited state, endowed with a transition
dipole moment, are taken into account. Generally speaking, excitons, both Frenkel
and Wannier types/l are neither Bosons not Fermions. Specifically , Frenkel excitons in such model systems are to be considered as Paclions or being equivalent
to S = l/2 spins which obey Bose-type and Fermi-type commutatio n relations with
respect to different and the same atomic sit~s, respectively."> This part-Boson,
part-Fermio n nature of relation, when taken as the deviation from the Bose-type
relation, is called the kinematical interaction. 6l· Furthermor e, there exist intrinsic
exciton-exci ton interactions , called dynaJUical interactions , which are to be taken
into account even if they are regarded as Paulions or, more generally, as electronhole pairs. Little discussion has been made on the nature of the dynamical interaction. These situations are rather familiar in, the field of Heisenberg 6l and
Ising7l magnets.
A Theory of Frenkel Excitons Using a Two-Level-Atom Model
1849
Historically, however, the difficulty associated with the kinematical as well
as dynamical interactions has, be.en by-passed in almost all studies of exciton problems by treating excitons simply as Bosons. 1l> 8l The underlying assumption is
that the exciton concentration is very smalL A brief discussion of the kinematical
interaction of Frenkel excitons has been made by Anderson. 5l A .more detailed
study has been made by Agranovich and Toshich9l by using a generalized form
of ,the Holstein-Primakoff transformation employed in the theory of magnetism. 10l
The discussion along this line is, however, generally much involved. It appears
that one of the points that are of primary importance in exciton problems is to
for~ulate a theory which is, at least formally, free from the· outset from the assumption of the exciton concentration being very small. Recent development of
laser physics has made it possible to produce high density excitons. Good progress has been made in experimentaF1 l and theoreticaF 2l studies of high density
W annier excitons in certain semiconductors.
On the other hand, in 1957 Feynman, Vernon and Hellwarth have studied
the behavior of a system of "non-interacting" two-level atoms in interaction with
an external electromagnetic field for the purpose of solving maser problems. 18l
These workers have shown that paying attention to Pauli operators themselves,
we can develop a simple but rigorous and geometrical -or, in a certain sense,
classical picture of an equation describing the atom-field interaction, which is
analogous to the Bloch equation in magnetic resonance,14l without resorting to
the conventional perturbation theory. Such a system has since been taken up as
a working model for the theoretical study of nonlinear optics 15l and laser or maser
physics. 16l
The principal purpose of this paper is to develop a rigorous, formal theory
of Frenkel excitons interacting with an external field, within the framework of
a two-level-atom model, which is, in its spirit, a generalization of the theory of
Feynman, Vernon and Hellwarth18l to the case of interacting two-level atoms.
The dipole approximation*l is employed here in treating the atom-field interaction,
the external field being taken as a c-number. Instead of treating excitons as a
non-ideal Bose gas, 9l the theory is formulated to describe Frenkel excitons in
terms of atomic polarization operators and level population operators, with particular attention paid to nonlinear Frenkel excitons. Very recently, Haken and
·his co-worker have studied the nonlinear optical properties of Frenkel excitons
using a method which appears to be somewhat different from the present one. 17l
It is shown that Frenkel excitons thus obtained are generally regarded as quantum-mechanical nonlinear polarization waves, the nonlinearity being characterized
by the level population. Although. Frenkel excitons as polarization waves have
been recognized for some time/ 8l' 19 l arguments presented so far do not appear to
*> This has recently been discussed by Stenholm, with references of several previous works
on this and related problems.44>
1850
S. Takeno and M; Mabuchi
be rigorous· compared to the present one. A by-product of the present theory
would be that we can get more intuitive and correct picture of Frenkel excitons,
which reduce, in certain limiting cases, to classical polarization waves derivable
from the Drude-Loren tz inodet2°l
The outline of this paper is as follows. In the next ·section an exact model
exciton Hamiltonian is derived within the framework of a two-level-at om model
adopted here. In § 3 atomic polarization ' and level population are in-troduced as
two fundamental quantities. In § 4 a pair of equations satisfied by these two
quantities are derived. Approximat ion procedures are employed in § 5 to treat
these two equations. In § 6 a brief discussion is given on nonlinear exciton-phot on
coupled modes. The last section is devoted to a briefsumma ry of results obtained
in this paper.
§ 2. A two-level-at om model of Frenkel excitons
We consider an assemblage of identical atoms or molecules, which may be
a molecular solid, a molecular aggregate or atoms or molecules embedded in a
solid matrix, in interaction with an external electromagn etic field.*l We study
collective electronic excited states in the atomic system. It is assumed at the
outset that the Heitler-Lon don scheme can be used for the electronic states of
the atomic system. in the zero-order approximati on. Let ef lmd (/)£f be the natural
energy eigenvalue and the correspondi ng eigenfunctio n of an atom at a site. i
specified by quantum number f. Also, let at1 and a£f be creation and annihilation operators of an electron in the state (/)£f for which the conventiona l anticommutation relations hold; any two a's belonging to different atomic sites are taken
to be commutable with each other. **l We take the Hamiltonian of the atomic
system to be of the form
H
= :E
efatfaif + (1/2) :E :E (ff'l V (ij) lf"f 111 )a£iajraif•a w
if
i j ff'f'f"
.
(2·1)
Here, V(ij) and V' (i) denote the Coulomb interaction between the i and j atoms
and the interaction of the i atom with the external field, respectively . The matrix elements of these quantities are defined by***l
(ff'l V(ij) lf"f 111 ) =
(fl V(i) If')=
*> The
for the sake
**> The
each other.
***> The
JJrpff (rt) rp"fr (r,) V(i}) rpw(rt) rpw(r,) dridr
Srpt, (rt) V' (i)
1 ,
(/)if'
(rt) drt,
(2·2)
(2·3)
constituents of the system may be atoms or molecules, which we hereafter call atoms
of simplicity.
electronic wavefunctions referring to different sites are assumed to be orthogonal to
exchange interaction is neglected. '
A Theory of Frenkel Excitons Using a Two-Level-Atom Model
1851
where ({J~(ri) is the complex conjugate of (/Ji1 (ri) and dri is the volume element
of an electron belonging to the i atom. Equation (2·1) without the last term
has been used previously by Dyub 21 ) in order to obtain approximate equations
describing Frenkel and W annier excitons by use of a double-time Green's function method. 22 )
In what follows we take into account only two of the energy states of a
given atom, one is the ground state and the other is an excited state, between
which there is a dipole matrix element. These two states are denoted by the
indices f = 0 and f = 1, respectively. It is as~umed here that the spectrum of the
applied field contains no appreciable ·component at any frequency which could
couple (/Ju and qJ;.o resonantly to other levels. Their effect could be found by
perturbation theory, and amounts, for almost all practical purposes, to small effective level shifts, which we suppose to have been included already in s1 and
s0• In terms of the a's, the exciton operators Ai + and Ai are defined by
(2·4)
and
where a';.= (6/", 6l, 6() with 6;.±=6/"±i6/ are the Pauli operators. Several relations satisfied by these quantities are listed below without proof for later use :6)' 16)
(2 · 5a)
(2·5b)
(2· 5c)
(2·5d)
(i=l= j)
(2·5e)
(2· 5f)
where nit defined by Eqs. (2·5a) and (2·5b) is the population of the f-th energy
level of the i atom and L1 (ij) is Kronecker's delta.
It is shown by carrying out somewhat lengthy though straightforward calcula6ons using Eqs. (2 · 5) that Eq. (2 ·1), when specialized to the system of twolevel atoms, reduces to*l
(2·6)
where
(2·7a)
*l
Any constant .factor appearing in expressions for the Hamiltonian will be omitted hereafter.'
1852
S. Takeno and M. Mabuch i
in which
D,= (1/2)~{(101 V(ij) J10)+ (01J V(ij) JOl)-2( 001 V(ij)'JOO)},
J
(2·7b)
and
J(ij) =(101 'V(ij) I01)=(0 1IV(ij) llO>=< OOJ V(ij) J11)=(1 1J V(ij) JOO),
(2·7c)
I(ij) = (111 V(ij) Jll)- (01J V(ij) J01)- (lOJ V(ij)l1 0) + (OOJ V(ij),JO
O),
(2·7d)
v,=(11 V(i) IO)
and
(2·7e)
The physica l meanin g of most of the quantiti es d~fined above is the same
as those
given, for example, by Davydo v. 1l That is, Di is the shift of the
natural line
frequen cy a>o = e1 - e0 of the i atom due to the presenc e· of the other
atoms, *l
J(ij) =J(ji) ·.is the dipole-dipole interact ion energy represe nting the
transfer energy of excitati on between the i and j atoms/ 8l I(ij) is the ,intrins ic
or dynamical interact ion energy between a pair of exciton s, and v, and v, +
describ e the
exciton-field interact ion. A detailed discussi on of I(ij) will he given
elsewhe re.
Equatio n (2 · 6), identica l in form with the Hamilto nian of a spin-1/2
Heisenb erg
magnet with anisotro pic exchang e interact ions, is rigorou s within the
framew ork
of the two-level-atom model adopted here. Several apprmd mate propert
ies of
element ary excitati ons associat ed with the model Hamilto nian (2 · 6)
with~ut the
last two terms have been studied previou sly by Anderso n. 5l Equatio
n (2·6) may
be general ized as follows :
H =
:Et a>o,li/ +
(1/2) :E J(ij) (a, +a1- + 111+a,-) + (1/2) :E J' (ij) (lit +a1+ + 11, -a -)
1
tJ
tJ
.
'
(2·6')
Here, the assump tion that J(ij) and J' (ij) may be differen t from each
other is
due to the inclusio n of the exchang e interact ion.
The convent ional procedu re in making a theoreti cal study of exciton problem
s
is to approxi mate the exciton operato rs A,+ and At by Bose operato
rs B, + ·and
B,, respecti vely, under the assump tion that the exciton density is very small
(cf.
Eq. (2 · 5f)). To take into account kinematic'al as well as dynami cal
interact ions
of excitons , Agrano vich and Toshich have introdu ced the followin g transfor
mation :9l
co
A,= [
:E
{( -2)"I (1 + v) !} (B,+)" (B,)"] f2B,
JI=O
1
co
and
At+=B ,+[ :E
{( -2)"/(1 +v)!} (B,+)"(B,)"Jf2,
v=O
which is a general ization of the Holstein -Primak off
*> We use units with A=l througho ut this paper.
tran~formation
(2·8)
in the theory
A Theory of Frenkel Excitons Using a Twa-Level-Atom Model
1853
of magnetism. 10l A stal'lcdwd prO'Cedme here would be to· transfo.Fm the Hamiltonian
{2·6) into the form
H="E, WoiBi+Bi+ (1/2)"E, J(ij) (Bi+B,+B,+Bi+Bi+B,++B,B,)
i
ij
(2·9)
where U(ij) is an effective interal!tion between Boson-like. excitons in which the
kinematical and dyl'lamical interactions have been incoTporated. From Eq. (2 · 9)
we could discuss several collective properties of Frenkel excitons as a non-ideal
Bose gas as suggested, for example, by Bogolitibov in the theory of superfluidity. 2'l
A treatment of the obtained Hamiltonian along this line is entirely omitted in
this paper.
§ 3. Atomic pelarization operators and level p·opnlation
An alternative expression for Eq. (2'· 6) written in terms of !J/, !J/, !J/ 1s
H= "'E. (J);!J;_' + 2 "E. J (ij) !J/"!5/' + (1/2) "E. I(ij) !I ;'!I/+ 2
i
ij
ij
"E.i V;!J/,
(3·1)
where
Wi = Wo; + (1/2) 'E. J(ij)
(3·2)
j
:and we have put vi=vi+· This is fo·rmal:ly equivalent to the so-called XY-model
Hamiltonian for Heisenberg ma.gnets. Aside from the case of magnetism, the
model Hamiltonians having tlle forms similar to Eq. (2 · 6) or (3 ·1) have been
used previously in the problem of lattice models .of liquid helium, 25 )' 26 ) ferroelectrics27l'26) and so on. The main difference of the model Hamiltonian (2 · 6) or
{3·1) from those employed in these cases is that the interactions J(ij) and I(ij)
.are of fairly long-range nature depending on the relative distance Ii- il as Ii- il-s
.and Ii- il-5, respectively::+:) In the specific case of one-dimensional systems with
only nearest neighbour interactions between a pair of atoms, exact expressions
for the eigenvalues of the Hamiltonian identical in form with Eq, (3 ·1) have
been obtained by Katsura~ 9 ) for magnetic systems and by Chesnut and Suna for
I
Frenkel excitons. 30 ) The physical properties of the model Frenkel-exciton Hamiltonian written in terms of Pauli operators have been studied extensively by LyndenBell and McConnell, 31 ) by K:rrugler, Montgomery and Mc:Connell, 82) and by Soos. 38>
These workers also limited their discussions to the case of one-dimensional systems with nearest neighb.our in:teractions, p,aying particular attention to the properties of specific organic crystals such as BDPA34l and TCNQ. 35l
The physical meaning of the Hamiltonian (3 · 1) can be understood more
dearly and intuitively by using the dipole approximation in describing the e?'citon*l
This point will be discussed elsewhere.
1854
S. Takeno and M .. Mabuch i
field interacti on, treating the external :field as .a c-numbe r:
v.=v,+= -!J.e,·E• or v,(O"i++O"i-)
=
-p.e.·E i with p.=2!J.O"/,
(3·3)
and by observin g the r~lation obtainab le from Eqs. (2 · 5a) and (2 · 5b):
(3·4)
In Eq. (3 · 3) !J. is the matrix element of the ·atomic dipole moment P• associate
d
with the i atom between the ground state ((J£o ·and the excited state q;u, e,
is a
unit vector in the direction of Pi, a~d Ei is an external :field seen by the i atom.
The quantity
(3·5)
is the. ·expectat ion value of Pi with respect to the quantized atomic wavefun
ction
'IJI"i(r.) defined by
(3·6)
Thus, as is well known in the case of non-inte racting two-leve l-atom models
extensively employe d in laser physics and nonlinea r optics, the quantity 0"/" and,
0"/
are intimatel y c?nnecte d with the atomic dipole moment p, and the populatio
n
differenc e n. defined by Eq. (3 · 4), respectiv ely. In terms of these two quantitie
s
and also of the populatio n of the excited level nil, Eq. (3 ~ 1) is rewritten
as
=Hex- I:; Piei · Ei
(3·7)
i
or
H
=
2: Woinil + (1/2) 2: T
i
ij
(ij) PiPJ + (1/2) 2: I (ij) nun11 - l:P•e• · Et .
ij
i
(3·7')
where
T(ij)
=
J (ij) I /1.2 ,
(3·8)
and the quantity Hex defined above is the intrinsic exciton Hamilton ian which
exists m the absence of the external field.
On the other hand, the correspo nding classical Hamilton ian that is obtained
from the Drude-L orentz mod.ef may be written .as .
H= (1/2) 2: M. (dui/dtY + (1/2) 2: Miwlul+ (1/2) 2: T(ij)p,p1 - l:Pi·E,
i
li
iJ
with
pi=e,ui ,
'
i
(3·9)
4
Theory of Frenkel Excitons Using a. Two-Level-Atom Model
1855
where u, is the .displaceme1;1t of an oscillator at a site i, M;, e, and (J); are its mass,
charge ,and eigenfrequency, respectively, and the meaning of T(ij) is the same
as that defined by Eq. (3 · 8). It is seen that the second and fourth terms in the
first line of Eq. (3 · 7) or (3 · 7') are identical with the · third and fourth terms
in Eq. (3 · 9), while the terms containing level populations ·have no direct classical
analog. Equations (3 ·1) and (3 ~ 7) or (3 · 7') describe Frenkel excitons as quantum-mechanical polarization waves. These are characterized by the atomic dipole
Pt and the level population difference n;.
§ 4. Equations of motion
Equations of motion for the Pauli operators rJ/', rJ;", rJ( are obtained from
Eq. (3 ·1) as follows:
dd/'/dt= -w;*rJl,
drJ;" /dt = w;*rJ/' -2v 1*rJ(,
(4·1c)
. drJ//dt=2v;*rJ/,
where
· w,* =wo+ :E I(ij) {rJ/+ (1/2)} =wo+ (1/2) :E I(ij) (n 1 + 1)
J
j
=wo+:E I(ij)n11
j
(4·2)
and
(4·3)
m which
(4·4)
with
It is now seen from Eq. (4·1a) that the physical meaning of rJl is intimately
connected with the quantity dp./dt. Equations (4·1) are summarized in the form
of a vector equation:
with
ll; = (2v;*, 0, w,*).
(4·5)
Thjs is exactly equivalent to the result obtained first by Feynman, Vernon and
Hellwarth in the case of non-interacting two-level atoms/ 8) provided w;* and v,*
a,re rep)aced by wo and v;, respectively, and also analogous to the Bloch equation
in magnetic resonance.14l Here, ());* at?-d v;* defined above can be thought of as
an effective excitation energy and an effective exciton-field interaction associated
with the i atom, respectively. The time evolution of these two quantities are
governed by the equations:
= 2 :E I(ij) v 1*rJl,
(4·6)
= -2 :EJ J(ij)w 1*rJ/.
(4·7)
dw,* /dt = (1/2) :E I(ij) (dn 1/ dt)
J
dv;*/dt=2 :E J(ij) (drJ//dt)
j
j
1856
S. Takeno- and M. Mabuch i
It is easily seen from Eqs. (4·2} and (4··3} and also from the reiation
J(ii) =l(ii)
=0· that both of (J),* and· 'vi:" ar'e commut able with any of (J/', (J/,
(J/, but they
are not commut able with each other;.
Equatio ns of motion obeyed by h ami
are readily obtained· from Eqs. (4 ·1)
as. follows:*>
n,
(4·8)
(4··9)
where
r,. ((J), *) =
.. * ((J), *)-1 •
- (j)'
(4·10)
These equatio ns are exactly equival ent to Eqs. (4·1a),.. ..,(4·1c} ,. but
-in._the·form
( 4 · 8) and ( 4 · 9) we have a simple physica l interpre tation. The atomic
dipole
momen t respond s to an applied· field
accordi ng to a driven' polariza tion-wa ve
equation , with the feature that the couplin g constan t - 2ti(J),*n, and
the term
-2;lw,* n, ~ 1 T(ij)p1 represe nting the exciton transfer from one atomic
site to
another are proport ional to the slowly varying level populat ion n,, reversin
g sign
when n, passes through zero. Equatio n ( 4 · 9) is simply a stateme nt
of conserv ati-on of energy. On the other hand-, from Eqs, (3·· 7) or (3 · 7') and
( 4 · 9) the
energy conserv ation of the total exciton system is express ed by the
equatio n
e,
(4·11)
There is one further immedi ate consequ ence of writing the equatio ns
in this form:
Multipl ying Eq. (4·8) by p, from the right, substitu ting Eq. (4·9)
and neglecting the time variatio n of (J)t* and th:e non-com mutativ ity of p, and
p,, we get
after integrat ing
(4·12)
where we have determi ned the constan t of integrat ion by reqmrm g
that (p 1)t=o
= (p,),=o = 0 and (n,),=o = ± 1. In the limiting case of non-inte
racting two-lev el
atoms in which (J)c*-+(J)o, Tt ((J)c*) -+0~ T(ij) -+0 and C:,*-+C:,, the equatio
ns obtained
above, in conjunc tion with the· Maxwe ll equation , hawe been used
as equatio ns
of fundam ental importa nce in the theory ofi laser physics and n<>nline
ar optics. 1fi),lB)
It is of interest to note· that the classica l counter part of Eq. ( 4·· 8) obta-inab
le'
from the Drud:eL orentz model is of the form
(4-13)
In Eq. ( 4 ·13) we have also included relaxati on effects through · a dampin
g con-·
stant Tt· On the other hand, any mechan ism giving rise to the
relaxati on of
*>'The time derivative of a quantity A will hereafter be. denoted- by
for the sake of simplicity.
A
wheneve r appropria te
A Theory of Frenkel Excitons Using a Two-Level-Atom Model
1857
electronic excitations to other degrees of freedom has not been taken into account
in our formulation of the problem. It is seen that Eq. ( 4 · 8) contains a resistive
force arising from the exciton-exciton interactions, which is either positive or
negative depending on the sign of wi*. An approximat~ expression for ri (a>t*)
is readily obtained by replacing Wi * by Wo and by using Eq. (4 · 6):
ri (wi*) = - (1/2wo) L; I(ij) n1 •
(4·14)
j
Equations ( 4 · 8) and ( 4 ·13) can be recast into the form*l
Pi+ r~Pi + Wi 2Pi + ai (0) wle- ;Jli) :E T (ij) Pi= ai (0) w/ (- ;Jli) ei '
(4·15)
j
where
for Eq. (4·8)
;Jl.= { ni
'
-1 for Eq. (4·13)
and
_ { 2;iw~l (w/- w2) for Eq. (4·8)
ai(w)2 M
2
2
ei / i (wi -w) for Eq. (4·13)
(4·16)
is the polarizability of the i atom. Thus, as mentioned before, the essential difference of Eq. (4 · 8) from Eq. ( 4 ·13) is the appearance of the terms involving
the level population which introduce into Eqs. (4 · 8) and ( 4 · 9) a nonlinearity;
this is of fundamental importance in the present case as well as in the case of
systems composed of non-interacting two-level atoms. Such systems have been
used extensively as a working model for the study of nonlinear optics and laser
or maser physics.
Besides the semi-classical dipole approximation in describing the exciton-field
interaction as given by Eq. (3 · 3), no approximation has been used in deriving
Eqs. ( 4 · 8) and ( 4 · 9) from Eq. (3 ·1). These equations are to be treated simultaneously with the Maxwell equation satisfied by the macroscopic electric field
E, which we take to be of the form
VE- (4n6./c (aE/at)- (lf..'e' /c WE/at
2
2)
2)
2)
=
(4nlf..' /c2)
(a 2P/at 2) .
(4·17)
Here, c and If..' are the light velocity and the magnetic permeability of the medium
respectively,~(}. is a conductivity that is introduced to include the effect of nonresonant losses in the medium, and
(4·18)
is the macroscopic polarization density in which r is the continuous space variable.
In Eq. ( 4 ·17) , the effect of all the energy levels other than the two levels under
consideration of a giv.en atom has been incorporated into a factor s', and the
volume of the system has been taken to be unity. A set of equation ( 4 · 8),
( 4 · 9), ( 4 ·15) and (4 ·16), when treated simultaneously, are generally highly nonlinear even in the limit of T(ij) ~o and I(ij) ~o, which cannot be treated without resorting to various approximation procedures.
*l
Here, w,* is rewritten as w, for Eq. (4·8) for the sake of simplicity.
1858
S. Takeno and M. Mabuchi
§ 5. Nonlinear excitons
In this section we make a brief study of the approximate properties of solutions of Eqs. (4 · 8) and ( 4 · 9). For this purpose we consider a specific case
in which the spatial arrangement of atoms in the system is periodic. We take
the electric field to be of the form
(5·1)
where the envelope functions e/±l=e/±) (t) are assumed to be slowly varying
in the sense that we/±> -:?e/±l and ket<±>-:? 17e/±>. In correspondence to this, we
decompose Pt into a positive and negative frequency parts:
Pt=fl.qt =fl. {q/-> exp (i8;)
+ q/+> exp ( -ie;)}.
(5·2)
Inserting Eq. (5 · 2) into Eq. ( 4 · 8), we get*>
where
Tt 1 (k)
= T(ij) exp [ik · (j- i)].
(5·4)
In obtaining this result we have omitted terms containing q)±> and q/±l which
appear in the evaluation of P• and r (w.*) p., respectively. An alternative expression for Eq. (5 · 3) derivable by using the relation qi+i_= exp (j · 17) q. is
±iq/"'>-
[w -w•*
(kY :r i_ t(w.*) +
2w
. 2
2
f1.
2
w.*n• ·{T(=rk-il7)- T(:rk)}
w
.
.J· q/"'>
(5· 3')
where
(5·5)
Here, the quantity
T(k)
=:EJ T. 1 (k) =:E T(ij)exp[ik·
J
(j--:i)] =T(-k)
·.
(5·6)
is the Fourier transform of the. dipole-dipole interaction energy or dipole sums,
a detailed calculation of which has been made by Cohen and Keffer. 36 > On the
other hand, Eq. (4 · 9), when combined with Eq. (5 · 2}, becomes
iw.*'lit = 2/}.w {q.<+>e.<->- q/->e.<+>}
+ 2fJ. 2w :E {Tt1 (k) q•Hq/+>- T.1 ( - k) q/+>q/->}
J
(5·7)
*> The subscript i attached to n(w,*) is omitted hereafter.
A Theory -of Frenkel Excitons Using a Two-Level-Atom Model
1859
or
iw;*n; = 2/f.w{q/+)ei(-)- qiHei<+>} + 2!J. 2 w {q;HT(k -il7)qi<+>- q;<+>T(- k- il7)q;H},
(5. 7')
where we have made use of a rotating-wave approximation, neglecting terms containing exp(±2i0;) and exp[±i(0;+01)]. Applying the same approximation procedure to Eq. ( 4 · 12) , we get
(5·8)
One thing easily seen from Eq. (5 · 3') is that w1 * (kY defined by Eq. (5 · 5),
when specialized to the limiting case
(5·9)
reduces to
(5·10)
The second of the limit (5 · 9) corresponds to the case in which the exciton density is vanishingly small, while the first one holds if the exciton-exciton interaction is negligible. Equation (5 ·10), first derived by Agranovich, 8l is normally
· · obtained from Eq. (2 · 9) with all the U's set equal to zero, by employing the
Bogoliubov transformation. The limiting case given by Eq. -(5 · 9) corresponds
to "linear" excitons · traditionally studied by many workers. The quantity w1* (k)
defined by Eq. (5 · 5) can be thought of as quasi-eigenfrequencies of "nonlinear"
excitons, which depend on the amplitude q/±l through the relation (5 · 8). It differs from the linear exciton eigenfrequencies w(k) in two respects; the one is
the shift or the modulation of the natural .excitation energy wo due to the excitonexciton interaction and the other is the modulation of the exciton energy band
which is characterized by the quantity - !J. 2n 1 T(k). The latter is either positive
or negative depending on whether n; is negative or positive and vanishes when
n 1 = 0. The last case corresponds to the equal population of the two energy levels.
Without detailed discussion we might expect that the properties of nonlinear excitons are very different· from those of conventional linear excitons when there
exists population inversion n 1 >0.
Another thing worth noticing in studying Eq. (5 · 3) or (5 · 3') is, as mentioned before, the existence of the imaginary part ± iwr (w 1*) in the eigenfrequencies,
which is to be added to Eq. (5 · 5). An approximate expression for r (w;*) as
well as w1* can be obtained by neglecting the space variation of n; (and therefore w1*) in treating Eqs. (4·2) and (4·14). Then, we obtain
wo* = wo + (1/2) (n + 1)1 and
r (wo*) =- ln/2wo,
(5·11)
where
I= I:: I(ij),
J
(5·12)
1860
S. Takeno and M. Mabuchi
and we have rewritten (J)i* and ni as (}) 0* and n, respectively . The effect of the ·
factor r ((}),*)could be seen by neglecting the term (ti(J),*n,j(})) {T(Tk-ifl )- T(Tk)}
in Eq. (5 · 3'), which amounts to assuming the space variation of q, to be negligible. A solution of Eq. (5 · 3') thus obtained may be written as
.
-
q/"l = exp(±iO/ -0") [ q,<"l (0) Ti(fl./(J)) .[(J)/"n£/"l exp(TiO' + O")dt'J.
(5·13)
where
and
and q, (0) is the value of q, at t = 0.
tained from Eq. ( 4 · 14) as follows :
0/' = - (1/4(}) 0)
0/'= (1/2)
rtr((J),*)dt', (5·14)
Jo
An approximate expression for
.E I(ij) (n1 + 1) = - (1/2(J) ),E I(ij)n 1
1,
0
j
j
'
0/~
is ob-
(5·15)
where we have put n, (0) = -1. According to a result of calculation of Eq. (2 · 7d)
using hydrogen-li ke atomic wavefunctio ns, which will --be given elsewhere, the
exciton-exci ton interaction energy I is shown to be positive. Thus, the sign of
the quantity r ((}),*)' which is determined by the population of the excited states
of the atomic system, is negative, giving rise to the amplificatio n of the polarization waves; The order of r ((}),*) is If (}) 0 , which is usually small since there exists
.a relation (}) 0 -:?;>l in almost all cases of physical interest.
§ 6. Nonlinear exciton-ph oton coupled modes
The 'equations for q, and -n, obtained in the previous section must be treated
:simultaneou sly with Eq. ( 4 · 15). In ~doing this we put
P=N0p=N0p{q<-l exp(i8) +q<+l exp( -i8)}=Nop q _
(6·1)
e=e<.-) exp(i8) +e<+l exp( -i8)
(6·2)
with
p=fl.e,
e=e·E
and
8=(J)t-k·r .
(6·3)
Here, N 0 is the number of atoms per unit volume in the ~ystem, and q<±land n
are taken to be func.tions of r, satisfying the equations:
±ii_/"l- [(})2 -(J)*(kY Ti_r((J) 0*) + fl. 2(J)o*n {T(Tk-if' ) -T(Tk)}]q <"l
2(})
2
(})
fl.(J)o *ne<">
(6·4)
(})
i(J)o*n =2fl.(J) {q<+le<-l- q<-le<+l} + 2112(}) {q<-lT(k- if') q<+)- qc+lT( -k-if') q<-l}
(6·5)
A Theory of Frenkel Excitons Using a Two-Level-Atom Model
1861
or
(6·6)
These are obtained from Eqs. (5 · 3'), (5 · 7') and (5 · 8) by omitting all the indices
i and then by taking the continuum liniit. Here, we have rewritten wi* as w0*
for the sake of convenience.*l An equation corresponding to Eq. (4·17) which
can be obtained by using the same procedure as that employed in proceeding
from Eqs. ( 4 · 8) and ( 4 · 9) to Eqs. (5 · 3), (5 · 3'), (5 · 7), (5 · 7'), (5 · 8) using
Eqs. (5 ·1) and (5 · 2) is written as
[(wl2) {1- (ele')} =Fi(2mr.le'.a')]e<'>l
=Fi (d*e<"'lldt)
+ (2nNo.awle')q<"'l ± i(4nNo.ale') q_<+J =0,
(6·7)
d* I dt = (a/at)
+ (el .a'Y12 (11e') c (k · V) I k ,
(6·8)
where
and the quantity defined by
e=e(w, k) =c2k 2la> 2tL'
(6·9)
is identified wit)l a frequency-dependent dielectric function of the system.
Equations (6 · 4), (6 · 5) or (6 · 6) and (6 · 7), when treated simultaneously,
are generally highly nonlinear, which may yield various types of exciton-photon
coupled modes. It is to be noted that conventional polariton modes 8J,IBJ here correspond to the specific case in which the space and time variation of q<±l, n and
e<±l are neglected in solving Eqs. (6 · 4) and (6 · 7) simultaneously. A result of
such a calculation is written as
(6·10)
Here,
(6·11)
are the plasma frequency, the oscillator strength and the zero-frequency polarizability
(See Eq. (4·16) .) of a two-level atom with energy separation w0* in isolation,
resp~ctively, in which· ·m is the electron mass. Equation (6 ·10) with imaginary
parts omitted is also rewritten as
(6·12)
where
(6·13)
*> See, also, Eq. (5·11).
1862
S. Takeno and M. Mabuchi
is identified with the squared energy eigenvalues of longitudinal excitons. Equation (6·12), which has a form similar to the Lyddane-Sa chs-Teller relation, 37>
determines, in conjunction with Eq, (6 · 9), the eigenfreque ncies of polariton modes.
The situation here is entirely analogous to the. case of phonons in ionic crystals, 38>
where w* (k) and WL* (k) are the eigenfreque ncies of transverse and longitudinal
optical phonons, respectively . Aside from the presence of the factor r (wo*), the
principal difference of the above result from conventiona l ones obtainable from
linear exciton theories is the appearance of level population n in the second terms
in the right-hand sides of Eqs. (6 ·10) and (6 ·13) which reverse sign_ when
passing through zero. Thus, the optical properties of the system, such as the
dielectric function e and the dispersion of polariton modes in the vicinity of
w = w* (k) ~ are expected to be qualitatively different when there exists population
inversion n>O, for which WL* (k) <w* (k).
The remaining part of this section is devoted to expressing Eqs. (6 · 4) ,..._,
(6 · 7) in a slightly different form. We take the electric field e and the polarization amplitude q (c£. Eq. (6·1)) to be of the form
e=e' cos 8,
(6·14)
q=q' cos 8-q" sin 8
(6·15)
with
8=wt-k·r+ ¢(r, t).
Here, the phase function ¢ (r, t) is introduced as a generalizati on of Eq. (6· 3),
which is also assumed to b,e slowly varying. The quantities q' and q" thus
defined can be identified as the electric dipole dispersion and absorption components, respectively . The relationship s among the amplitudes or the envelope functions introduced in the previous and present cases are
(6·17)
q' = q<+> + q<->'
q" = i (q<+>- q<->).
(6·18)
A set of equations satisfied by q', q" and e' as well as n can be obtained by
putting Eqs. (6 ·16), (6 ·17) and (6 ·18) into Eqs. (6 · 4) ,..._, (6 · 7). After somewhat
lengthy though straightforw ard calculations , we get*>
q' = (Aw + ¢) q"- (r/2) q' + nvex (k) · r q!- !!__ __!___: , q",
2
(6·19a)
mex
q" = - (Aw + ¢) q'- Cr /2) q" + Jl.ne + nvex (k) . , q" + !!__ __!___: , q''
2 mex
1i = - Jl.eq"- (q'vex (k) · rq" + q" Vex (k) · rq')
.
+ l_ __!___:
2
(6·19b)
(q'"q"- q"" q'), -
mex
(6·20)
*> The quantity C:' is hereafter rewritten as
e for
the sake of simplicity.
A Theory of Frenkel Excitons Using a Two-Level- Atom Model
1863
(6·21)
f:¢+ (w/2) {1- (c/c')}f:+a q'=O,
(6·22a)
+ (2n6./s' p') - aq" = 0,'
(6·22b)
(d*f:/ dt)
where
Jw=w-w*( k)
and
a=2nNo!-!W/c'.
(6·23)
The quantities Vex (k) and mex are the velocity of a linear exciton with momentum
k and the effective ma~s tensor of the exciton, respectively , which are defined
by the equations
(6·24)
where w< 0l (k) is defined by Eq. (5 ·10). Equations (6 ·19), (6 · 20), (6 · 21), (6. 22)
correspond to Eqs. (6 · 4), (6 · 5), (6 · 6), (6 · 7), respectively . Here, several approximation procedures have been employed to arrive at analytically tractable
forms. First of all, we have employed an approximati on T (~ k- ir) = T (k ± ir)
=T(k) ±i(VkT)V· - (1/2) crkrkT), and we have set (w 2 -w*(k))/2w and Wo*/w
and w< 0l (k) / Wo equal to Jw and unity, respectively . The first two approximati ons
can be used if we are concerned with the frequency in the vicinity of w= Wo *.
A similar approximati on has often been employed in the theory of nonlinear
optics. The last approximati on can be used if the exciton band width is very
small compared to Wo or w0*. In Eqs. (6·22a) and (6·22b) the term (4nN0p/s')
(8/at) has been omitted, which is small compared with a. Incidentally , it is
shown by applying the same argument as that used in deriving Eq. (4·12) that
Eqs. (6 ·19) with the factor r omitted ~nd (6 · 20) satisfy Eq. (6 · 21). Equations
(6 ·19) ~ (6· 22) can be. considered as a generalizati on of the results obtained by
McCall and Hahn, 39 ) by Lamb, and by others 15l in the field of nonlinear optics
to the case of Frenkel excitons. No general solution of these equations has been
found even in the limit of vanishing interatomic interactions . These workers
have studied several nonlinear optical properties, such as self-induced transparenc y,
superradian ce,· etc., using non-interact ing two-level-at om models. Very recently,
a multi-soliton solution of a specific form of Eqs. (6 · 19) ,.__, (6 · 22) in this limit
has been noted. 40 ) It may be of interest if Eqs. (6 ·19) ~ (6 · 22) yield soliton-like
exciton-ph~ton coupled modes. It is expected that if we take into account the
in treating Eqs. (6·4)~(6·7) or (6·19)~
time and space variation of q,n anD.
(6 · 22), we could obtain exciton-pho ton coupled modes whose properties are different from those of conventiona l polariton modes. It is shown that under certain
circumstanc es a particular solution of a specific form of Eqs. (6 ·19) ~ (6 · 22) can
be expressed in terms ef elliptic functions. These and other problems related
to solutions of the equations derived in the previous and this sections are worth
,separate discussions and will be studied elsewhere.
e
1864
S. Takeno and M. Mabuchi
§ 7.
Concludi ng remarks
It has been shown in this paper that within the framewor k of a two-level
atom model Frenkel excitons are rigorously expressed in terms of the XY-mode l
Hamiltoni an (3 ·1). As is well known in the theory of magn~tism, there are
several ways to treat the Heisenber g model Hamiltoni an. In this paper we have
confined ourselves to the study of equations of motion satisfied by Pauli spin
operators. Equations ( 4 · 8) and ( 4 · 9) are the fundamen tal kinetic equations thus
obtained which describe nonlinear Frenkel excitons in terms of two basic quantities ,
atomic polarizatio n operator and population ·difference .. As an applicatio n, the
properties of nonlinear Frenkel excitons and nonlinear exciton-ph oton coupled
modes have been studied in §§ 5 and 6. The arguments presented therein were
rather preliminar y, and a more detailed study of these problem ~ill be given
elsewhere . If the population of the excited state of a given atom is negligibly
small, Eq. (4 · 8) gives results essentially equivalent to those obtained from con- ·
ventional linear exciton theories. Aside from jts possible applicabil ity to exciton ·
problems in general, Eq. ( 4 · 8) or (4 ·15) may also be useful to calculate the
van der Waals force by observing the fact that it can be recognized as the decrease in the sum of the zero-point energies. of Frenkel excitons upon formation
of atomic or molecular condensat e. 5l' 41l
In this paper statistical- mechanica l approach to Eq. (3 ·1), which has been
conventio nally employed in the theory of magnetism , has been entirely. omitted.
From the known properties of the Heisenber g model Hamiltonia n·, we can infer
the thermodyn amical or statistical- mechanica l properties of Frenkel excitons .. One
thing easily seen without any detailed calculation is the possible existence of the
phase transition, We can expect that the appearanc e of ·magnetic ordering in
magnetic system may probably be related to the occurrenc e of sliperradia ilt states
first discussed by Dicke. 42l The situation here may be similar to the ca~e of a
lattice model of liquid helium discussed by Matsubara and Matsuda. 25 l
The results obtained in this paper may be generalize d to several directions .
One of the possibiliti es is to include relaxation effects in Eqs. ( 4 ·1} in a manner
entirely analogous to the case of the Bloch equation in magnetic resonance ; and
the other is to generalize a two-level- atom model adopted here. In its most naive
form the later may be done by assuming that each atom consists of several atomic
dipoles or oscillators whose natural eigenfrequ encies are equal to the energies
of the excited states allowed by dipole transition. Another point worth noticing
is to take into account the effect of configurat ion interaction s giving rise to collective exciton fusion or fission. 43l
References
1)
A. S. Davydov, Theory of Molecular Excitons,. translated by S. B. Dresner (Plenum Press,
New York-Londo n, 1971).
A Theor.y of Frenkel Excitons Using a Two-Level-Atom Model
1865
2) ·See, for example, M. T. Vala, J. R. Silbey, S. A. Rice and J. Jortner, J. Chern. Phys. 41
(1964), 2846.
3) See, for example, W. Rhodes, ]. Am. Chern. Soc. 83 (1961), 3609.
4) R. S. Knox, Solid State Physics, edited by F. Seitz and D. Turnbull, Suppl. 5 (Academic
Press, 1963).
5) P. W. Anderson, Concepts in Solids (W. A. Benjamin, Inc., New Y ark-Amsterdam, 1963),
p. 132.
6) F. ]. Dyson, Phys. Rev. 102 (1956), 1217.
7) See,. for example, T. D. Schultz, D. C. Mattis and E. H. Lieb, Rev. Mod. Phys. 36 (1964),
122.
8) V. M. Agranovich, Zh. Eksp. i Teor. Fiz. 37 (1960), 430 [English transl: Soviet Phys.JETP 10 (1960), 307].
9) V. M. Agranovich and B. S. Toshich, Zh. Eksp. i Teor. Fiz. 53 (1967), 149 [English
transl : Soviet Phys.-JETP 26 (1968), 104].
10) T. Holstein and H. Primakoff, Phys. Rev. 58 (1940), 1098.
11) H. Kuroda, S. Shionoya, H. Saito and E. Hanamura, Solid State Commun. 12 (1973), 533
and ·references cited ther~in.
12) See, for exampe, E. Hanamura and M. Inoue Proceedings of the International Conference
on the Physics of Semiconductors (Warsaw, 1972), p. 711 and references cited therein.
13) R. P. Feynman, F. L. Vernon and R. W. Hellwarth, J. Appl. Phys. 28 (1957), 49.
14) F. Bloch, Pliys. Rev. 70 (1946), 460.
15) G. L. Lamb, Rev. Mod. Phys. 43 (1971), 99 and references cited therein.
16) H. Haken, Laser Theory, Handbuch der Physik 25/2C (Springer-Verlag, Berlin, 1970) and
references cited therein.
17) H. Haken and A. Schenzle, Phys. Letters 41A (1972), 94; Z. Phys. 258 (1973), 231.
18) ]. ]. Hopfield, Phys. Rev. 112 (1958), 1555.
19) W. Rhodes and M. Chase, Rev. Mod. Phys. 39 (1967). 94.
20) See, for example, F. Wooten, Optical Properties of Solids (Academic Press, New York
and London, 1972), Chap. 3.
21) I. P. Dyub, Zh. Exp. i Teor. Fiz. 39 (1960), 610 [English transl: Soviet Phys.-JETP 12
(1961), 429].
22) D. N. Zubarev, Uspekhi Fiz. Nauk 71 (1960), 71 [English transl: Soviet Phys.-Uspekhi 3
(1960), 320].
~
23) W. R. Heller arid A. Marcus, Phys. Rev. 84 (1951), 809.
24) N. N. Bogoliubov, J. Phys; USSR 11 (1947), 23.
25) T. Matsubara and H. Matsuda, Prog. Theor. Phys. 16 (1956), 569.
26) R. T. Whitlock and P. R. Zilsel, Phys. Rev. 131 (1963), 2409.
27) M. ~Tokunaga and T. Matsubara, Prog. Theor. Phys. 35 (1966), 581.
28) R. Blinc, J. Phys. Chern. Solids 13 (1960), 204.
29) S. Katsura, Phys. Rev. 127 (1962), 1508.
30) D. B. Chesnut and A. Suna, J. Chern. Phys. 39 (1963), 146.
31) R. M. Lynden-Bell and H. M. McConnell, J. Chern. Phys. 37 (1962), 794.
32) ]. I. Krugler, C. G. Montgomery and H. M. McConnell, ]. Chern. Phys. 41 ·(1964), 2421.
33) Z. G. Soos, J. Chern. Phys. 43 (1965), 1121.
34) See, for example, A. S .. Edelstein, J. Chern. Phys. 40 (1964), 488.
35) See, for example, D. B. Chesnut and W. D. Phillips, J. Chern. Phys. 35 (1961), 1002.
36) M. H. Cohen and F. ~Keffer, Phys. Rev .. 99 (1955), 1128.
37) See, for example, C. Kittel; Introduction to Solid State Physics, 4th edition (John Wiley
and Sons,. Inc., New York, London, Sydney, Toronto, 1971), p. 184.
38) M. Born and K. Huang, ·Dynamical Theory of Crystal Lattices (Clarendon Press, Oxford,
1954).
39) S. L. McCall and E. L. Hahn, Phys. Rev. 183 (1969), 457.
1866
S. Takeno and M. Mabuchi
P. J. Caudrey, ]. D. Gibbon, ]. C. Eilbeck and R. K. Bullough, Phys. Rev. Letters 30
(1973), 237.
41) D. Langbein. J. Phys. Chern. Solids 32 (1971), 133 and references cited therein.
42) R. H. Dicke, Phys. Rev. 93 (1954), 99.
43) See, for example, Y. Tomkiewicz , R. P. Groff and P. Avakian, ]. Chern. Phys. 54 (1971),
4504 and references cited therein.
44) Stenholm, Phys. Letters C6 (1973), 1.
40)