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On the problem of non-equilibrium population of excitons
Physica 95A (1979) 359-366 © North-Holland Publishing Co.
ON THE PROBLEM OF NON-EQUILIBRIUM P O P U L A T I O N OF EXCITONS R. MAKSIMOVIC Home O~ice, High School, Belgrade, Yugoslavia
M. MARINKOVIC Boris Kidri~ Institute of Nuclear Sciences, Belgrade, Yugoslavia
a. TO~IC Institute of Physics, University of Novi Sad, Yugoslavia
Received 13 April 1978
The non-linear response of the exciton system is considered and non-equilibrium exciton populations are calculated. It is shown that in the high frequency domain (to > toc~ 10-~3s-t) the role and the influence of the external perturbation on the exciton population becomes unimportant.
1. Introduction Due to the high energies of excitons, which are of the order of 104 kB, the increase of the temperature of the crystal cannot cause spectacular effects. See for example refs. 1 and 2. In the mentioned papers the mean values of the exciton population number were calculated over the equilibrium ensemble and the thermodynamical characteristics of the exciton system were analysed by the use of these equilibrium mean values. We shall consider here the corresponding non-equilibrium ones, i.e. the mean values of a type (~+0D#)n.~q. taken over a non-equilibrium ensemble. The operators OD+ and 9a# create and annihilate excitons of the type a and/3, respectively. The interaction Hamiltonian of the type ~D, will be treated as a perturbation, ~ being the external periodical electric field, and D - t h e total dipole m o m e n t of the exciton system. To simplify the calculations we assume that the interaction is introduced adiabatically, i.e., the lower limit of the S matrix of the system, to tends to -o0. The use of a finite S matrix limit would not lead to difficulties, but 359
360
R. MAKSIMOVIt~ et al.
makes the calculations c u m b e r s o m e . The s y s t e m of excitons corresponding to a multi-level molecular excitations scheme will be analysed. The effects of D a v y d o v splitting will not be taken into account as the crystal lattice is considered to be simple, i.e. with one molecule per e l e m e n t a r y cell.
2. Non-linear response of the exciton system The Hamiltonian of the exciton system, corresponding to the multi-level molecular excitations scheme, can be written as follows: £ Ho = Hg + ~
Y~u,~,(r
-
r ') ~ (+ r)~,(r)~(r
(1.1)
÷ ' ) ~ , ( r ').
~ , v E ( 1 , 2 . . . . . r). Here: Hg = [Eo + ½U0o0o]V, X ~ ( r - r') = [(E~ - Eo - U0ooo)a~ + ½(U~o0 + Uoo~)]a(r - r')
+ ~[~b.oo~(r - r') + $o~.o(r - r')], 1
I
r
Y ~ , ~ , ( r - r') = ~{~b~,~,(r - r') + ~[~b~,0o(r - r') + C#oo~,(r - r ) ] ~ , + ~boooo(r - r')Su.,8.~,}.
The exciton operators ~ . ( r ) and ~ ( r ) are constructed on the electron o p e r a t o r s a . ( r ) and a+u(r) in the following way ~ . ( r ) = a ~ ( r ) a . ( r ) and ~+~(r) = a+~(r)ao(r). T h e y satisfy very complicated c o m m u t a t i o n rules (see ref. 3) and are called the quasi-Pauli operators. If the concentration of exciton operators is low, the quasi-Pauli operators can be a p p r o x i m a t e l y replaced by Bose operators B . ( r ) and B+~(r). The indices V-, /x', v and v' denote the type of the molecular excitations and " 0 " denotes the ground state of the molecule. The energies E . are the eigen-values of the Hamiltonian of an isolated molecule, i.e. H(~r)$u(~r)= E~Ab~(~r), !~r being the set of the internal coordinates of the molecule placed at r and the $~,,,~,(r - r') are the matrix elements of the operator of the dipole-dipole interactions: ~,vv,( r
I
r') = f d3~r d3~r, ~l*~(~r)~*v(~r,)A (r - r ' ) ~ v ' ( ~ r ' ) ~ ' ( ~ r ) ,
A (r- r')= ~
e2
t ~.~r'- 3 [ ( r -
r')~r][(r]r -- r'] 2
r')~r,]~J
NON-EQUILIBRIUMPOPULATIONOF EXCITONS
361
The volume of the crystal is denoted by V, e is the charge of the dipole and Um,,,,~,= f d3r ~b~,~,(r). The center of the symmetry of the crystal coincides with the inversion center of the molecule so that the third order products in exciton operators do not appear in the Hamiltonian. Moreover the terms proportional to ~ and ~+~+ are neglected as their contribution may be considered as small4). The interaction Hamiltonian is given by Hint(t)
=
- -
f d3rg(rt)D(r),
where
D(r,= ~ [ ~ ' ~ ( r ) 3
~,~ = ~,,,
+ ~o~(r) + ~ ~.~(r)~(r)],
+
~,,, = 0
and
g(rt)
f
J d3k do) [$(k, o)) e ikr-i~t -~- c.c.],
$(k, 60) = $ ( - k , 6o).
In the interaction representation we can write: ¢¢(t) = e -~°t/iA/qint(t) e ~°t/ih = ff'(l)(t) + l~a)(t),
¢¢(')(t)= - ~ / d3r d3k doJ[$(k, oo) eikr-iot+ c . c . ] [ ~ 0 ~ ( r t ) + c.c.], ~/(2)(t) = - - ~ f d3r dak dto[s(k, aJ)eikr-iot +
(1.2)
c.c.]~v~ +~(rt )~u( rt ).
We shall calculate the following non-equilibrium mean value
(~+~(rt)~o(rt))n.e¢ = (S-'(t, to)~+~(rt)~a(rt)$(t, to))e¢, a,/3 •(1,2 . . . . . F), where
t
t0)=: exp[ f d, #(t)]. to
(1.3)
362
R. MAKSIMOVI(~et al.
In the last formula 7~ is Dyson's chronological operator, F0 is the free energy of the unperturbed system and 0 = k B T is the temperature in energy units. Expanding the S matrix up to the quadratic terms in the interaction, we obtain (~+.(rt)~(rt))..eq. =(~'+.(rt)~.(rt))eq + ~
dt(~+(rt)~(rt)
~¢V'~2)(t)
to
tO
to
-idtlidt2{~+(rt)~(rt)l~/(l)(tl)~/(2)(t2) to -
tO
~/¢'")(t2)1~'"l(tt)~+.(rt)~(rt)).q.
(1.4)
After decouplings of the type (~l(Xt)~iz(X2)~(X3)~il(X4))eq
= °'~ i2/1(X2- Xt)~i4i3(X4-
-}- o'~i4i1(X4- X t ) I i 2 i 3 ( X 2
X3) --
XI),
where ~ , i ( x - y) = ( ~ ' ( y ) ~ , ( x ) ) ~ . . x =- ( r t ) .
I,i(x - y) = ( ~ , ( x ) ~ ( y ) ) e q ,
y =- (r't').
are correlation functions, and after going over to the Fourier components A i ~ ( x _ y ) = f dqAij(q)eiqt~
r),
q---(k,w),
qx=kr-wt,
to --*-do.
We finally obtain:
e;~ (k, ,o) = N~(t,, ,,,) + N~(k, ,,,) + X~(k, ,,,),
- -(2~r)' ~ T ~ f a3, dOx(,, O) ×
(1.5)
[@o~-~8~,l..(q, O) + f~t~o.~..(q,O)]},
JV'~(k, ~o) = i(2~')4
2, ~ f d'eda~.~(k,,o)
× [Ia.(q, 12)~.~(q - k, 12 - ~ ) - I.~(q - k, 12 - w)~C~.(q, 12)1,
(2~r)~ ~, f × [5.~(q - it, 12 - ~o)~.(q, 12) - ~ . . ( q - k, 12 - w)I~.(q, 12)
NON-EQUILIBRIUM POPULATION OF EXCITONS
363
x I ~ ( q - k, 12 - ~o) - Ia~(q, 12)offu~( q - k, 12 - to) - I~(q
- k , 12 - o ~ ) ~ ( q ,
12)]},
where < ~ + ~ ) =- ( ~ : ( r t ) ~ a ( r t ) ) e q , x ( k , to) = 2Is(k, to)12+ s2(k, t o ) + s*2(k, tO), L ( k , to) = s(k, to) + s*(k, to), r ( k ' , to', k", to") = [s(k', to')s*(k", to") + s ( k ' to')s(k", to") + c.c.],
(1.6)
.N'~(k, o9) = (2"rr) -4 f d3r d t ( ~ : ( r , t)~a(r, t))neq e -itr+i.,t.
As we see the term ¢¢a)(t) of the interaction Hamiltonian contributes to N~(k, to) in linear approximation and leads to dissipative processes in the system. The first non-zero contribution of ff'(1)(t) is in the quadratic approximation. So we can conclude that the external interaction (1.2) causes the dissipative processes in the system and also leads to the space-time dispersion in mean values of the type ( ~ : ~ ) . As it is well known, the equilibrium mean values (~:~a)eq do not depend on k and w.
3. Non-equilibrium mean values of the exciton occupation number
We proceed with the analysis of the quantity Na,(k, a,) using the harmonic approximation for excitons, i.e. in the Hamiltonian (1.1) we neglect the term proportional to Y and we replace the quasi-Pauli operators ~ and ~+ by the Bose operators B and B ÷. Since the correlation functions, figuring in (1.5) can be expressed through the corresponding Green's functions, we first calculate the Green's functions G~°)(r - r', t - t') = O ( t - t')(Bi(rt)B-~(r't') - Bj(r't')Bi(rt))~q, O(t_t,)={lo
for t > t', for t < t',
(2.1)
B i ( r t ) = e -~°°>t/ihBi(r) e/~°alt/ih,
for the harmonic Hamiltonian n~0O)= ~ ~" d3r d3r , x~(r p,v J
r')B+~(r)n~(r').
(2.2)
364
R. MAKSIMOVI(~ et al.
_ , ~ . _ r', t - t') can be written as follows The s y s t e m of e q u a t i o n s for •rz.(0)~_ ih O--G~)~(r - r', t - t') = i h S ( r - r ' ) 6 ( t - t')6.~ at
~,~. .... + ~ f d3r,,Xo.(r_ r ,,,~(o), a, f l E ( 1 , 2 . . . . .
r' , t - t'),
(2.3)
F).
A f t e r the F o u r i e r t r a n s f o r m a t i o n s .~. G(O)t.
-- r ' ,
t
--
t') =
f
d3k dw ,_,~(°)ll'.~,to) elk(,-, ') i,o(t c~,
(2.4)
X . ~ ( r - r') = f d3k X . ~ ( k ) e ikr '
the s y s t e m of e q u a t i o n s (2.3) can be written in matrix f o r m
[~ - )~(k)l(~(o)(k, to) : ~ih i ,
(2.5)
where
G(°)(k, to) = [IG~A(k, to)ll, k --Ilhto~.A, ~(k)
= IIX.,(k, to)ll,
] = II~-,ll.
I n t r o d u c i n g the unitary matrix hT/(k) which diagonalizes the matrix X ( k ) , i.e.:
~'(k)M(k) = M(k)~(k),
~(k) = Ilhf.o(k)a~ll,
(2.6)
we obtain G ( ~ ( k , to) = i h [ / ~ _ ~ ( k ) ] - ' = M ~(k)G(°)(k, to)M(k) = [IG..(k,
21r
to)~.~ll, (2.7)
i 1 G ~ ( k , w ) = 2zr to - f . ~ ( k ) + i6"
As we see, the h a r m o n i c G r e e n ' s f u n c t i o n of the exciton s y s t e m is diagonal, so that the c o r r e s p o n d i n g correlation f u n c t i o n s , a c c o r d i n g to the general relations (see ref. 5, p. 216) 3~#(k, t o ) = [ G i j ( k , t o + i S ) - G i j ( k , Iij(k, to) = e ~1° 5~o(k, to),
n(to, O) = [e ~ / 0 are diagonal too.
1] -1,
to-i6)]rl(to,
O),
8-~0, (2.8)
NON-EQUILIBRIUM POPULATION OF EXCITONS
365
C o m b i n i n g (2.7) and (2.8) with (1.5) we find N ~ ( k , to) in the h a r m o n i c approximation
[.Ar~(k, to)](o) = (B+B,~)eqS,,aS(k)8(to) x{1
~(2~r)S~'~f_ j
-
I
0,1'X[4,
/..(q)]cotgh~}, (2.9a)
-
,
2
n
exp [h[~o(4-----))]]
× n [/z~(4), O]n[f~,,(q - k ) , 0]/8{to - [ / z z ( q ) - f ~ ( 4 - k)]}, 3
(2.9b)
(3) co)](°) = ~(2~r) 8 f d3q ~oa~'/3"r[q - k, f~,,~(q [,N~,,(k,
-
k); q, f/3/3(q)]
X{e ~r~'~/° ~ [ / ~ ( q ) , 0] - ~ [ / ~ ( 4 - k), 0]
- 2 e ~r~°-~l° n [ / ~ ( 4 ) , 0] x ~ [/~a(4 - k), 0118{~o - [faa(4) -f,~,~(4 - k)]}.
(2.9c)
Finally, we can find the non-equilibrium exciton o c c u p a t i o n n u m b e r . Putting a = / 3 in eqs. (2.9) w e obtain the following e x p r e s s i o n f o r the nonequilibrium e x c i t o n o c c u p a t i o n n u m b e r at 0 ~ 0: ,1~0)o _ (21r) s f
× 8{to - [/~(q)-fa,~(q
-
k)]}.
(2.10)
In o r d e r to e s t i m a t e the b e h a v i o u r of [ N ~ ( k , to)](e°~0 w e shall take app r o x i m a t e l y r [ q - k, f , ~ ( q - k); q , f ~ ( q ) ] = ~ ' ( ~ ) , ~ = h-t(E~ - E0). T h e effective m a s s a p p r o x i m a t i o n will be used too, so that we can write hk
f,~,~(q)-fa,~(q - k) = 2m----~(2q cos A - k),
AE(0,~'),
q / > k _+ Q_( k ) 2
Q(k)-
'
2m~to hk '
w h e r e m~ is the effective e x c i t o n m a s s of the z o n e a. With the a b o v e m e n t i o n e d a p p r o x i m a t i o n s w e finally obtain D¢'~ (k, oo)](o°~o= (2--~-~ ]m~[ ]fl~o~[2r(~)
-
-
.
(2.11)
As it is seen, the non-equilibrium e x c i t o n o c c u p a t i o n n u m b e r (calculated per unit square v o l u m e and unit square time) increases c o n t i n u o u s l y w h e n ,o ~ 0.
366
R. MAKSIMOVIt~ et al.
a n d t e n d s to z e r o w h e n to ~ t o ~ ) ( k ) w h e r e to ~)(k) = 2-~(2qmax - k).
(2.12)
It is a l s o s e e n t h a t ~f,~ i n c r e a s e s w i t h d e c r e a s i n g k. T h e v a n i s h i n g o f ~f,, at to = to~ c a n be, c o n d i t i o n a l l y , c o n s i d e r e d as a t r a n s i t i o n f r o m t h e " s t i m u l a t e d " to t h e " u n s t i m u l a t e d " p h a s e , b e c a u s e f o r to > toc t h e i n f l u e n c e o f t h e e x t e r n a l s t i m u l a t i o n *~'(t) to t h e e x c i t o n p o p u l a t i o n n u m b e r s b e c o m e s negligible a n d t h o s e c h a n g e s a r e d u e to t h e t h e r m a l effects, mainly.
4. Conclusion The results of the analysis of the non-linear response of the exciton system c a n b e s u m m a r i z e d as f o l l o w s : a) T h e e x t e r n a l p e r t u r b a t i o n l e a d s to t h e s p a c e - t i m e d i s p e r s i o n * o f t h e m e a n v a l u e s o f t h e t y p e (3~+~)eq. It c a n a l s o c a u s e t h e d i s s i p a t i v e p r o c e s s e s in t h e system of excitons. b) T h e n o n - e q u i l i b r i u m e x c i t o n o c c u p a t i o n n u m b e r at l o w t e m p e r a t u r e s d e c r e a s e s w h e n t h e f r e q u e n c y a n d t h e w a v e v e c t o r i n c r e a s e . It v a n i s h e s at t h e f r e q u e n c y to~ a n d this m e a n s t h a t f o r to > to~ t h e r o l e a n d t h e i n f l u e n c e o f the external stimulation on the exciton populations becomes unimportant. A c c o r d i n g to t h e f o r m u l a (2.12), t a k i n g qmax ~ 10Scm ~ a n d k ~ 105cm ~ w e find t h a t this f r e q u e n c y limit is o f t h e o r d e r tOc -~ l0 ~3s -~ i.e., it is t w o o r d e r s o f m a g n i t u d e l o w e r t h a n t h e f r e q u e n c y o f light.
References 1) 2) 3) 4) 5)
R. Diordjevi6 and B.S. Togi6, Progress of Theor. Phys. 54 (1975) 1299. M. Marinkovi6, Phys. Stat. Sol. (b) 69 (1975) 291. D.J. Lalovi6, B.S. Toni6 and R.B. Zakula, Phys. Rev. 178 (1969) 1472. V.M. Agranovich, Zh. Ekspi. Teor. Fiz. 37 (1959) 430. S.V. Tyablikov, The Methods of Quantum Theor. of Magnetism, (Nauka, Moscow, 1975) p. 216 (in Russian).
* In contradistinction to the equilibrium mean value, the non-equilibrium mean value depends on k and to.
ON THE PROBLEM OF NON-EQUILIBRIUM P O P U L A T I O N OF EXCITONS R. MAKSIMOVIC Home O~ice, High School, Belgrade, Yugoslavia
M. MARINKOVIC Boris Kidri~ Institute of Nuclear Sciences, Belgrade, Yugoslavia
a. TO~IC Institute of Physics, University of Novi Sad, Yugoslavia
Received 13 April 1978
The non-linear response of the exciton system is considered and non-equilibrium exciton populations are calculated. It is shown that in the high frequency domain (to > toc~ 10-~3s-t) the role and the influence of the external perturbation on the exciton population becomes unimportant.
1. Introduction Due to the high energies of excitons, which are of the order of 104 kB, the increase of the temperature of the crystal cannot cause spectacular effects. See for example refs. 1 and 2. In the mentioned papers the mean values of the exciton population number were calculated over the equilibrium ensemble and the thermodynamical characteristics of the exciton system were analysed by the use of these equilibrium mean values. We shall consider here the corresponding non-equilibrium ones, i.e. the mean values of a type (~+0D#)n.~q. taken over a non-equilibrium ensemble. The operators OD+ and 9a# create and annihilate excitons of the type a and/3, respectively. The interaction Hamiltonian of the type ~D, will be treated as a perturbation, ~ being the external periodical electric field, and D - t h e total dipole m o m e n t of the exciton system. To simplify the calculations we assume that the interaction is introduced adiabatically, i.e., the lower limit of the S matrix of the system, to tends to -o0. The use of a finite S matrix limit would not lead to difficulties, but 359
360
R. MAKSIMOVIt~ et al.
makes the calculations c u m b e r s o m e . The s y s t e m of excitons corresponding to a multi-level molecular excitations scheme will be analysed. The effects of D a v y d o v splitting will not be taken into account as the crystal lattice is considered to be simple, i.e. with one molecule per e l e m e n t a r y cell.
2. Non-linear response of the exciton system The Hamiltonian of the exciton system, corresponding to the multi-level molecular excitations scheme, can be written as follows: £ Ho = Hg + ~
Y~u,~,(r
-
r ') ~ (+ r)~,(r)~(r
(1.1)
÷ ' ) ~ , ( r ').
~ , v E ( 1 , 2 . . . . . r). Here: Hg = [Eo + ½U0o0o]V, X ~ ( r - r') = [(E~ - Eo - U0ooo)a~ + ½(U~o0 + Uoo~)]a(r - r')
+ ~[~b.oo~(r - r') + $o~.o(r - r')], 1
I
r
Y ~ , ~ , ( r - r') = ~{~b~,~,(r - r') + ~[~b~,0o(r - r') + C#oo~,(r - r ) ] ~ , + ~boooo(r - r')Su.,8.~,}.
The exciton operators ~ . ( r ) and ~ ( r ) are constructed on the electron o p e r a t o r s a . ( r ) and a+u(r) in the following way ~ . ( r ) = a ~ ( r ) a . ( r ) and ~+~(r) = a+~(r)ao(r). T h e y satisfy very complicated c o m m u t a t i o n rules (see ref. 3) and are called the quasi-Pauli operators. If the concentration of exciton operators is low, the quasi-Pauli operators can be a p p r o x i m a t e l y replaced by Bose operators B . ( r ) and B+~(r). The indices V-, /x', v and v' denote the type of the molecular excitations and " 0 " denotes the ground state of the molecule. The energies E . are the eigen-values of the Hamiltonian of an isolated molecule, i.e. H(~r)$u(~r)= E~Ab~(~r), !~r being the set of the internal coordinates of the molecule placed at r and the $~,,,~,(r - r') are the matrix elements of the operator of the dipole-dipole interactions: ~,vv,( r
I
r') = f d3~r d3~r, ~l*~(~r)~*v(~r,)A (r - r ' ) ~ v ' ( ~ r ' ) ~ ' ( ~ r ) ,
A (r- r')= ~
e2
t ~.~r'- 3 [ ( r -
r')~r][(r]r -- r'] 2
r')~r,]~J
NON-EQUILIBRIUMPOPULATIONOF EXCITONS
361
The volume of the crystal is denoted by V, e is the charge of the dipole and Um,,,,~,= f d3r ~b~,~,(r). The center of the symmetry of the crystal coincides with the inversion center of the molecule so that the third order products in exciton operators do not appear in the Hamiltonian. Moreover the terms proportional to ~ and ~+~+ are neglected as their contribution may be considered as small4). The interaction Hamiltonian is given by Hint(t)
=
- -
f d3rg(rt)D(r),
where
D(r,= ~ [ ~ ' ~ ( r ) 3
~,~ = ~,,,
+ ~o~(r) + ~ ~.~(r)~(r)],
+
~,,, = 0
and
g(rt)
f
J d3k do) [$(k, o)) e ikr-i~t -~- c.c.],
$(k, 60) = $ ( - k , 6o).
In the interaction representation we can write: ¢¢(t) = e -~°t/iA/qint(t) e ~°t/ih = ff'(l)(t) + l~a)(t),
¢¢(')(t)= - ~ / d3r d3k doJ[$(k, oo) eikr-iot+ c . c . ] [ ~ 0 ~ ( r t ) + c.c.], ~/(2)(t) = - - ~ f d3r dak dto[s(k, aJ)eikr-iot +
(1.2)
c.c.]~v~ +~(rt )~u( rt ).
We shall calculate the following non-equilibrium mean value
(~+~(rt)~o(rt))n.e¢ = (S-'(t, to)~+~(rt)~a(rt)$(t, to))e¢, a,/3 •(1,2 . . . . . F), where
t
t0)=: exp[ f d, #(t)]. to
(1.3)
362
R. MAKSIMOVI(~et al.
In the last formula 7~ is Dyson's chronological operator, F0 is the free energy of the unperturbed system and 0 = k B T is the temperature in energy units. Expanding the S matrix up to the quadratic terms in the interaction, we obtain (~+.(rt)~(rt))..eq. =(~'+.(rt)~.(rt))eq + ~
dt(~+(rt)~(rt)
~¢V'~2)(t)
to
tO
to
-idtlidt2{~+(rt)~(rt)l~/(l)(tl)~/(2)(t2) to -
tO
~/¢'")(t2)1~'"l(tt)~+.(rt)~(rt)).q.
(1.4)
After decouplings of the type (~l(Xt)~iz(X2)~(X3)~il(X4))eq
= °'~ i2/1(X2- Xt)~i4i3(X4-
-}- o'~i4i1(X4- X t ) I i 2 i 3 ( X 2
X3) --
XI),
where ~ , i ( x - y) = ( ~ ' ( y ) ~ , ( x ) ) ~ . . x =- ( r t ) .
I,i(x - y) = ( ~ , ( x ) ~ ( y ) ) e q ,
y =- (r't').
are correlation functions, and after going over to the Fourier components A i ~ ( x _ y ) = f dqAij(q)eiqt~
r),
q---(k,w),
qx=kr-wt,
to --*-do.
We finally obtain:
e;~ (k, ,o) = N~(t,, ,,,) + N~(k, ,,,) + X~(k, ,,,),
- -(2~r)' ~ T ~ f a3, dOx(,, O) ×
(1.5)
[@o~-~8~,l..(q, O) + f~t~o.~..(q,O)]},
JV'~(k, ~o) = i(2~')4
2, ~ f d'eda~.~(k,,o)
× [Ia.(q, 12)~.~(q - k, 12 - ~ ) - I.~(q - k, 12 - w)~C~.(q, 12)1,
(2~r)~ ~, f × [5.~(q - it, 12 - ~o)~.(q, 12) - ~ . . ( q - k, 12 - w)I~.(q, 12)
NON-EQUILIBRIUM POPULATION OF EXCITONS
363
x I ~ ( q - k, 12 - ~o) - Ia~(q, 12)offu~( q - k, 12 - to) - I~(q
- k , 12 - o ~ ) ~ ( q ,
12)]},
where < ~ + ~ ) =- ( ~ : ( r t ) ~ a ( r t ) ) e q , x ( k , to) = 2Is(k, to)12+ s2(k, t o ) + s*2(k, tO), L ( k , to) = s(k, to) + s*(k, to), r ( k ' , to', k", to") = [s(k', to')s*(k", to") + s ( k ' to')s(k", to") + c.c.],
(1.6)
.N'~(k, o9) = (2"rr) -4 f d3r d t ( ~ : ( r , t)~a(r, t))neq e -itr+i.,t.
As we see the term ¢¢a)(t) of the interaction Hamiltonian contributes to N~(k, to) in linear approximation and leads to dissipative processes in the system. The first non-zero contribution of ff'(1)(t) is in the quadratic approximation. So we can conclude that the external interaction (1.2) causes the dissipative processes in the system and also leads to the space-time dispersion in mean values of the type ( ~ : ~ ) . As it is well known, the equilibrium mean values (~:~a)eq do not depend on k and w.
3. Non-equilibrium mean values of the exciton occupation number
We proceed with the analysis of the quantity Na,(k, a,) using the harmonic approximation for excitons, i.e. in the Hamiltonian (1.1) we neglect the term proportional to Y and we replace the quasi-Pauli operators ~ and ~+ by the Bose operators B and B ÷. Since the correlation functions, figuring in (1.5) can be expressed through the corresponding Green's functions, we first calculate the Green's functions G~°)(r - r', t - t') = O ( t - t')(Bi(rt)B-~(r't') - Bj(r't')Bi(rt))~q, O(t_t,)={lo
for t > t', for t < t',
(2.1)
B i ( r t ) = e -~°°>t/ihBi(r) e/~°alt/ih,
for the harmonic Hamiltonian n~0O)= ~ ~" d3r d3r , x~(r p,v J
r')B+~(r)n~(r').
(2.2)
364
R. MAKSIMOVI(~ et al.
_ , ~ . _ r', t - t') can be written as follows The s y s t e m of e q u a t i o n s for •rz.(0)~_ ih O--G~)~(r - r', t - t') = i h S ( r - r ' ) 6 ( t - t')6.~ at
~,~. .... + ~ f d3r,,Xo.(r_ r ,,,~(o), a, f l E ( 1 , 2 . . . . .
r' , t - t'),
(2.3)
F).
A f t e r the F o u r i e r t r a n s f o r m a t i o n s .~. G(O)t.
-- r ' ,
t
--
t') =
f
d3k dw ,_,~(°)ll'.~,to) elk(,-, ') i,o(t c~,
(2.4)
X . ~ ( r - r') = f d3k X . ~ ( k ) e ikr '
the s y s t e m of e q u a t i o n s (2.3) can be written in matrix f o r m
[~ - )~(k)l(~(o)(k, to) : ~ih i ,
(2.5)
where
G(°)(k, to) = [IG~A(k, to)ll, k --Ilhto~.A, ~(k)
= IIX.,(k, to)ll,
] = II~-,ll.
I n t r o d u c i n g the unitary matrix hT/(k) which diagonalizes the matrix X ( k ) , i.e.:
~'(k)M(k) = M(k)~(k),
~(k) = Ilhf.o(k)a~ll,
(2.6)
we obtain G ( ~ ( k , to) = i h [ / ~ _ ~ ( k ) ] - ' = M ~(k)G(°)(k, to)M(k) = [IG..(k,
21r
to)~.~ll, (2.7)
i 1 G ~ ( k , w ) = 2zr to - f . ~ ( k ) + i6"
As we see, the h a r m o n i c G r e e n ' s f u n c t i o n of the exciton s y s t e m is diagonal, so that the c o r r e s p o n d i n g correlation f u n c t i o n s , a c c o r d i n g to the general relations (see ref. 5, p. 216) 3~#(k, t o ) = [ G i j ( k , t o + i S ) - G i j ( k , Iij(k, to) = e ~1° 5~o(k, to),
n(to, O) = [e ~ / 0 are diagonal too.
1] -1,
to-i6)]rl(to,
O),
8-~0, (2.8)
NON-EQUILIBRIUM POPULATION OF EXCITONS
365
C o m b i n i n g (2.7) and (2.8) with (1.5) we find N ~ ( k , to) in the h a r m o n i c approximation
[.Ar~(k, to)](o) = (B+B,~)eqS,,aS(k)8(to) x{1
~(2~r)S~'~f_ j
-
I
0,1'X[4,
/..(q)]cotgh~}, (2.9a)
-
,
2
n
exp [h[~o(4-----))]]
× n [/z~(4), O]n[f~,,(q - k ) , 0]/8{to - [ / z z ( q ) - f ~ ( 4 - k)]}, 3
(2.9b)
(3) co)](°) = ~(2~r) 8 f d3q ~oa~'/3"r[q - k, f~,,~(q [,N~,,(k,
-
k); q, f/3/3(q)]
X{e ~r~'~/° ~ [ / ~ ( q ) , 0] - ~ [ / ~ ( 4 - k), 0]
- 2 e ~r~°-~l° n [ / ~ ( 4 ) , 0] x ~ [/~a(4 - k), 0118{~o - [faa(4) -f,~,~(4 - k)]}.
(2.9c)
Finally, we can find the non-equilibrium exciton o c c u p a t i o n n u m b e r . Putting a = / 3 in eqs. (2.9) w e obtain the following e x p r e s s i o n f o r the nonequilibrium e x c i t o n o c c u p a t i o n n u m b e r at 0 ~ 0: ,1~0)o _ (21r) s f
× 8{to - [/~(q)-fa,~(q
-
k)]}.
(2.10)
In o r d e r to e s t i m a t e the b e h a v i o u r of [ N ~ ( k , to)](e°~0 w e shall take app r o x i m a t e l y r [ q - k, f , ~ ( q - k); q , f ~ ( q ) ] = ~ ' ( ~ ) , ~ = h-t(E~ - E0). T h e effective m a s s a p p r o x i m a t i o n will be used too, so that we can write hk
f,~,~(q)-fa,~(q - k) = 2m----~(2q cos A - k),
AE(0,~'),
q / > k _+ Q_( k ) 2
Q(k)-
'
2m~to hk '
w h e r e m~ is the effective e x c i t o n m a s s of the z o n e a. With the a b o v e m e n t i o n e d a p p r o x i m a t i o n s w e finally obtain D¢'~ (k, oo)](o°~o= (2--~-~ ]m~[ ]fl~o~[2r(~)
-
-
.
(2.11)
As it is seen, the non-equilibrium e x c i t o n o c c u p a t i o n n u m b e r (calculated per unit square v o l u m e and unit square time) increases c o n t i n u o u s l y w h e n ,o ~ 0.
366
R. MAKSIMOVIt~ et al.
a n d t e n d s to z e r o w h e n to ~ t o ~ ) ( k ) w h e r e to ~)(k) = 2-~(2qmax - k).
(2.12)
It is a l s o s e e n t h a t ~f,~ i n c r e a s e s w i t h d e c r e a s i n g k. T h e v a n i s h i n g o f ~f,, at to = to~ c a n be, c o n d i t i o n a l l y , c o n s i d e r e d as a t r a n s i t i o n f r o m t h e " s t i m u l a t e d " to t h e " u n s t i m u l a t e d " p h a s e , b e c a u s e f o r to > toc t h e i n f l u e n c e o f t h e e x t e r n a l s t i m u l a t i o n *~'(t) to t h e e x c i t o n p o p u l a t i o n n u m b e r s b e c o m e s negligible a n d t h o s e c h a n g e s a r e d u e to t h e t h e r m a l effects, mainly.
4. Conclusion The results of the analysis of the non-linear response of the exciton system c a n b e s u m m a r i z e d as f o l l o w s : a) T h e e x t e r n a l p e r t u r b a t i o n l e a d s to t h e s p a c e - t i m e d i s p e r s i o n * o f t h e m e a n v a l u e s o f t h e t y p e (3~+~)eq. It c a n a l s o c a u s e t h e d i s s i p a t i v e p r o c e s s e s in t h e system of excitons. b) T h e n o n - e q u i l i b r i u m e x c i t o n o c c u p a t i o n n u m b e r at l o w t e m p e r a t u r e s d e c r e a s e s w h e n t h e f r e q u e n c y a n d t h e w a v e v e c t o r i n c r e a s e . It v a n i s h e s at t h e f r e q u e n c y to~ a n d this m e a n s t h a t f o r to > to~ t h e r o l e a n d t h e i n f l u e n c e o f the external stimulation on the exciton populations becomes unimportant. A c c o r d i n g to t h e f o r m u l a (2.12), t a k i n g qmax ~ 10Scm ~ a n d k ~ 105cm ~ w e find t h a t this f r e q u e n c y limit is o f t h e o r d e r tOc -~ l0 ~3s -~ i.e., it is t w o o r d e r s o f m a g n i t u d e l o w e r t h a n t h e f r e q u e n c y o f light.
References 1) 2) 3) 4) 5)
R. Diordjevi6 and B.S. Togi6, Progress of Theor. Phys. 54 (1975) 1299. M. Marinkovi6, Phys. Stat. Sol. (b) 69 (1975) 291. D.J. Lalovi6, B.S. Toni6 and R.B. Zakula, Phys. Rev. 178 (1969) 1472. V.M. Agranovich, Zh. Ekspi. Teor. Fiz. 37 (1959) 430. S.V. Tyablikov, The Methods of Quantum Theor. of Magnetism, (Nauka, Moscow, 1975) p. 216 (in Russian).
* In contradistinction to the equilibrium mean value, the non-equilibrium mean value depends on k and to.