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Trap controlled hopping in one-dimensional systems
Sohd State Communications, Vol. 53, No. 5, pp. 4 5 1 - 4 5 5 , 1985. Prmted in Great Britain
0 0 3 8 - 1 0 9 8 / 8 5 $3.00 + 00 Pergamon Press Ltd.
TRAP CONTROLLED HOPPING IN ONE-DIMENSIONAL SYSTEMS V.N PrIgodln, Chr. Smdel* and A P Nakhmedov A.F. Ioffe Physlco-Technlcal I n m t u t , Academy of Sciences of the USSR, 194021 Leningrad, USSR
(Received 14August 1984 by A Zawadowsk 0 The results of a theoretical study of trap controlled hopping m 1D systems by means of the Berezlnsky techmque are presented. In the case of a discrete trap state we gwe exact time and field dependences of the relaxation current produced by rejected carriers. In an m~tlal time interval where the moving carriers are very effectively trapped one obtains exactly the result formerly found for the randomly broken bond model. Taking into account a small transverse hopping rate the 1D results are generahzed for quasl-lD systems The physical sense of the results Is d~scussed RECENTLY THERE IS high interest In the experimental study of orgamc materials (molecular crystals, polymers) being characterized by sufficient amsotropy and showing unusual transport properties [ 1 - 3 ] Various theoretical approaches have been used to mterprete the experimental data on the basis of hopping transport in disordered systems [ 4 - 8 ] Considerable success In the study of one-dlmensmnal (1 D) and quasi-1D systems, m partmular of thmr kinetic properties, has been achmved [ 6 - 8 ] using a special dmgram technique suggested by Berezlnsky [9] In the following we present the results of an exact study of the so-called trap controlled hopping m 1D systems and given a generahzatlon to the quasM D case The master equations governmg the charge carrier dynamms in the system under consideration reads
where w being an effective hopping rate without field and Vo is of the order of characteristic phonon frequencies u+ are u_ are capture and release rates of the trap whmh are assumed to be of the form u = 60exp (--et/kT),
u+ = co,
(3)
where e t is the trap depth and co is also of the order of characteristic phonon frequencies The total propagator Dnn'(t) IS given by
Dnn'(t) = Dnn'(t) + Dnn'(t)
(4)
with/)nn,(t ) = (fin(t)), Dnn'(t) = (Pn(t)) describing the transmon into a transport or trap state, respectively, at site n In Laplace representation
Dnn,(S ) = ; e-StDnn,(t)dt
(5)
0
dt
=
E (Wn,n+ fin+ --Wn÷ ,nfin)--U÷Pn
the current is determined by
g=+- 1
+u--P~
dL dt
](s) = eneas Z (n--n')Dnn'(s)
(6)
n
- u+fin--u--Pn,
(1)
where fin and *Pn are the conditional probablhtles to find at time t a moving carrier at site n in a "transport" state or in a trap state, respectively, if at t = 0 it was located in a "transport" state at site n', l.e, the boundary condition is fin(t = O) = 8nn' The transltmn rates wn, n+g depend explicitly on the apphed field E via the parameter 0 = eEa/2kT
wn+_qn - we = we +-°,
w = VOe-2a/rB
(2)
* Permanent address Instltut fur Polymerenchemle "Erich Correns", Akademle der Wlssenschaften der DDR, DDR - 1530 Teltow - Seehof, GDR 451
For solving (1) we use the Berezlnsky technique [9] The application of this method to the hopping problem in 1D systems has been described in detail [6, 10] Following the main ideas of this approach the interaction of moving carriers with randomly distributed traps is treated perturbatxonally The Green function Onn, is represented as hne from site n' to site n at which, similarly to the conventional cross technique, vertices are accomodated. Any diagram for Onn, can be decomposed into three blocks With respect to the external vertices at n and n' one has the left, the right and the central path Each of the blocks may be separately summed up Thus, consldermg blocks with a determined number of pairs of incoming and outgoing hnes, one obtains the following equations for the sum of
452
TRAP CONTROLLED HOPPING
right-part Rm(n ), left-part Lm,(n') and central-part diagrams K m,m(n', n )
Rm(n) = (z+/z-)-mnRm,
Lm,(n') = (z+/z_}m'n'Lm ,
Lm - Rm
(7)
Vol 53, No 5
scattering amplitude becomes f = -- 1, x e , the moving careers are trapped but there IS no release In agreement with (9) and (11), at f = -- 1 we have b Wmm, = (1 -- C)6mm, "t-C(-- 1)m~im,O
pVmm, = ( 1 - - C ) 6 m m , (z+lz_)mRm = [~ ~ Wmm'Rm'
(8)
m'
In this case we are able immediately to solve equations
(8), (10) and get
(.k~rn-k-lF2k+m'-rn ( .f)2 (m-/~) Wmm' = E ~ m ~ m ' - I J 1+ k vii
t
(9)
c ( - - 1) m Rm =
r
(z./z_) Km,m(n ,n) = D ~. Vmm"Km'm"(n ,n-- 1)
(z+/z-) m -- (1 -- c)
Km,m(n',r/) =
m"
[(1 --c)(z_/z+) m]n-n'-2(Z_/Z+)m~mm , at n > n'
at n > n'
Vmm'
(177
(10)
+f)2rn-2k+1
= E ~ ( 'm k t~~mk + ' m'-mj2k+m'-rn(1
k
(11)
Thus, equation (16) yields __
(~)m[( l - - c ) (z~)ml
~-'
~)nn,(S) = 1 f
n-n¢
zn_n,_
(12)
r = [(s + w÷ + w_) 2 - 4w_w÷]a'2
(13)
*
(z+ t
where
z._ = (s + w÷ * w_ +_r)/2w_
(18)
• (R,. + Rm÷l) Rm +~_gm.l
(191
Similarly we find
Together with the boundary conditions Ro = Lo =
1,
Km'm(n-- l,n)
= (Z_/Z +)mC~mm ,
L)nn'(S) = C E (-- l) mzn--n' (1 --C) $ m
(14) equations ( 7 ) - ( 1 1 ) form a complete set The quantity f i n (9) and (11) is the total scattering amplitude o f a moving carrier at a single trap
x (R m + Rm+l)
(20)
Equations (19), (20) are written for the case n > n', quite similar expressions are obtained at n < n' Hence the total current consists of two parts
H+/r f=
1 + u+/r + u_/s
(15)
The symbol/3 means averaging over all the possible trap configurations At n < n' one obtains slmdar expressmns for the central part. Combining the blocks and taking into account the various comblnauons with external vertices at n and n' one obtains
l(s) = ]free(s) +]tray(s)
(21)
with /tree(S) = 1°( 1 - - c ) 2 s
W
?"
u -6m -'(u 2 -
~m
1) 2 [1 - (1 - c ) : u - 4 m - 2 ]
[1 - - ( 1 --c)u-2m]2[1 - - ( 1 - - c ) 2 u - 2 m - 2 ] 2 *
z_n-n'(z+/z_) m÷ m'(R m, 4. Rm'+l) *
Dnn,(s ) = 1/r
I
rn, m t
• [e-°u -- (1 -- c)u-2m] 2 [e-°u -- (1 -- c)u-2m] 2 (22)
. K m , m ( n - - l n ) (, R m + ~ _ ) ( 1 6_) R' m + l . ]tr.p(s)
Similarly, the Green f u n c t i o n / ) . n ' can be written an terms of the three blocks As a first apphcatlon o f the basxs equations obtained above we discuss the case o f a discrete trap state occurring with concentratmn c, 1 e , in equations ( 8 ) - ( 1 1 ) with probablhty c we have u÷ 4 : 0 and with probablhty (1 -- c) u+ = 0 Following (15) in the highfrequency range given by the c o n d m o n u_/u÷ ~ s/r the
= 1o w
(1 -
c)c 2
E U-4rn+l(u2-- I)[1 - - ( 1 --c)2u -4m-2] m
[1--(1--c)u -2ml[1-(1-c)u
•
-2m-2]
1 * [ue_O
(l_c)u_2m ] [ueO_( l_c)u_2m]2,
(23)
where u 2 = z+/z_ and 1o = 2eneawshO is the current of a
Vol. 53, No. 5
TRAP CONTROLLED HOPPING
regular chain without disorder (traps). lfree corresponds to transmons where no trap state appears at site n, /trap to transmons into a trap state at n Summing up the two contnbutmns of the current one obtains precisely the expression known from the study of the randomly broken bond model [6, 7] This result is qmte surprising since we are concerned w~th two different physical problems described by different Green functions Their first moments, however, turn out to be equwalent. The same expression for the current is found if one starts from the representation
l(s) = 1o Y[ &,,(s)
(24)
?l
Equation (24) shows that only electrons occupying a "transport" state contribute to the current After carrying out the reverse Laplace transformation, in the weak field case (1 - c ) e ° < 1 one finds the following asymptotm time dependence at t ~ oo
l(t)/lo ~ exp [-- 2wt(ch ° -- 1)]
(25)
Since the current asymptotlcs contains no trap parameter, it is expected to occur due to field effects arising at the ends of clusters, I.e, m the nmghbourhood of traps. In the strong field case (1 --c)e ° > 1 large clusters gwe the mare contribution to the current asymptotms which is found to be 1
453
We note that the current asymptotlcs shows unusual behaviour for O/c -->0
j(t)/lo ~ exp { - - ( ~ )
'/3 [Tr(1--roal/2rar)/212/3}.(29)
In the field range 0 > c (c "~ 1,0 "~ 1) at t ~ ~o the current I (t) behaves as
\ ra, i - rd, ]
In the opposite frequency limit s/r ,~ u_/u+ the scattering amplitude f tends to zero. Then the sum in the equatlons for transfermatrlces Vmm,, W.,m, (9) and (11) can he restricted to terms where foccurs In the first order and one obtains Wmm'
= 8ram'
+ mf(6m',m+l +¢~m',m-1 + 28re'm) (31)
Vmm' = [1 + (2m + 1)f] 8ram' + (m + 1)f6m',m+l + mf6m,, m -*
(32)
It can be shown that expression (24) IS not related to a restricted frequency range but has general nature in the model under consideration [ 11]. Thus, we have
j(s) = ]oC,(q = O,s),
(33)
where G(q, s) is the Fourier transform of/3nn,(s )
[ ( 1 - - c ) 2 e 2 ° - - 112
c(1- - 3 t -o-F v - i]
j(t)//o
ex [-cw (o0
e-0
26,
The transition regime before reactung the asymptotms is more transparent m the hmit of low concentrations c "~ 1. For weak fields 0 ,~ 1 and 0 < e, one obtains
e-'q(n-n')Dnn,(s ) = O+(q)
+ D_(-- q) -- Doo with
Z+ Z-
D0o = 1/r E(g+lz-)m(Rm+Rm*l) R m + - - R m * l
(35)
1'2
+
I J \r~l
O±(q,s) = 1/r ~, Qm(q,s)(z+/z_) m
• exp --
(34)
*
[l_]_(~7"d-ifXel2]12t__~
r~
~
I'1= - ~
r¢l
2rrI/2xo
1(0/1o =
G(q, s) =
r ~ t - - 3 ( r r / 2 ) 2/3
X (1 --1 ;~)2 ~ ] 1/3}
m R=
Z+
+Rm+x - - , g-
(36)
(27)
-where D+ and D_ in (34) and (36) correspond to the cases n > n' or n < n', respectively In (36) we Introduced the auxiliary funcUon Q~(q, s) related to the original blocks by
where 1 ~'chf =
"'"
cZw '
1 "fdr --
2cw ° '
Xc
=
[
[ 1(
2t.__]rat~_ l ''3
Qm(q, ~ e-'qnQ~n( - 1,n) * s) = n=o
2 r~]]
Qm(q,s) = £ e-'onQm(-n, 1) .=o
(28)
(37)
454
TRAP CONTROLLED HOPPING
w~th
Vol. 53, No. 5
Finally for the current we get
Q~n(- 1, n)
=
1
z n_ [z+]m' I~_) (Rm'-t-Rrn'+l)
~,
× K'm'm(-- 1, n),
n > 0
_
Q~n(n, 1) = Zm, z+n [-~-) K,nm,(n, 1)(Rm, + Rm'+l)
(38)
n<0.
From (10), we obtain the following equation for a~n(q, s)
+ ~±(z+/z-) m * (Rm + Rm+~)
ch~o = 1 + 2u_/cu+
aim = ( - l)me-m~O(1 -- e - ~ Q o , 2u_ 1 Qo = cu+ sh~o (w/s)l/2
1
(39)
(40)
w± = Uo exp (-- 2/31r:); < w±
Due to the exit term m (44) there occurs a renormallzatlon of the bare Green funcUon Dnn, and of the parameters z±. For weak field, thin renormahzatlon gives effectwely an addmonal contribution to the field parameter 0, I e.,
(41)
The lncommmg transverse term is perturbatlonally treated Finally the Green function can be represented as
1 G(q, qi, s, 0") = G-:(q, s, 0") -- w±(q±)'
(50)
where G(q, s, 0") is the exact 1D Green funcUon (with corngated 0) and the Fourier transformed transverse hopping rate m
to the diffusion coefficient being equal to Do = a2w. Hence, the characteristic relation u_/cu+ of the trap parameters can be estimated by means of (42). At weak fields 0 < 1 in the equations for Rm, Q~ in (40), (41) instead of ~0there occurs the function ff given by
ch~ = ch~ + 2 U_w02/s
(44)
C U+
and in (41) Qo must be replaced by + 0
(45)
Q~ ~
wl(q: ) = w I ~ e'qXRu.
+0
(51)
In this way the current in the quam-lD system is found to be 1
101D/lo = 1 --ZWl(S, 0")/1o llD(s' 0")/Io
(52)
i.e., the asymptotic behavlour obtained above is maintained. As a result of the nonzero transverse hopping rate the field parameter 0 is changed only This effect can be understood as a change m the mean drift time
1/2"
C U+ S
(49)
W
(43)
-- --
(48)
(42)
/do = e/kTDo
sh~ ~ - -
(47)
where a l # are chain Indices The sum is extended over z± nearest-nelghbour chains The mtercham hopping rate w± is supposed to be much smaller than the lntercham ones
02 ~ 0 *2 = 02 + ziw----'~z
where Oo =/done Is related by the Einstein relation
2 u_ w CU+$
Conmdermg the cases Owls .~ 1,0 w/s >> 1 one obtains current asymptotlcs similar to (42) The results dmcussed so far can be generalized for a quam-1D system m which a small amount of mterchaln hopping is allowed In this case the master equation (1) must be completed by the term
I.t
I.e., at 0 = 0 we have Q~ = Q~n- The asymptotic conductwRy wtuch can be similar to (33) written in terms of G(q = 0, s), becomes
o(s)/oo = 1/s ~ , CU+ 1-t-u_
-- 1
s h ' - ~ (OF + Qo - 1) (46)
wz ~. Pu(s, u) -- ziwiPd(sl a),
with ~_+ = (z~) ~ 1 In agreement with (33), for calculating the current not the whole Q-function is reqmred but only Q~m(q = O) = Q~. Let us discuss several cases Without field, 1.e at 0 = 0, equaUons (8), (38) yield the following soluuon for R m, a ~
Rm = (-- 1)me ~m~°,
ch~
](s)/lo = 2w(s/w + 02) 1/2
Tdr
Tdr
T,.,r
T
TRAP CONTROLLED HOPPING
Vol. 53, No 5
with r± = w_~~. A similar result was estabhshed m the case of a continuous trap-depth distribution [12] In the field-free case the present result indicates different long-time behaviour of the diffusion constant for 1D and quasl-lD systems, respectwely.
5. 6. 7 8.
REFERENCES 1. 2.
3. 4.
D. Haarer & H. Mohrwald, Phys. Rev. Lett. 32 (1975) 1447. U. Selferheld, H. Bassler & B. Movaghar, Phys Rev Let. 51 (1983) 813 S. Klvelson, Phys. Rev Let. 46 (1981) 1344. J. Bernasconi, S. Alexander, W.R. Schneider & R. Orbach, Rev. Mod. Phys. 53 (1981) 175.
9. 10 11. 12.
455
G. Pfister & H. Scher,Adv. Phys. 27 (1978) 747. V.N. Prigodin & A.N. Samukhm, Preprmt A.F. Ioffe Physico-Techmcal Instltut Leningrad 1983823. V.N. Pngodm & A.N. Samukhm, Solid State Commun. 46 (1983) 379. V.N Prigodm, Zh. Eksp. Teor. Ftz. 86 (1983) 242. V.L Berezmsky,Zh. Eksp. Teor. Ftz. 65 (1973) 1251 V.N Pngodm, Zh. Eksp Teor. Fiz 82 (1982) 1221 V.N. Prigodin,Zh. Eksp Teor. Ftz. to be pubhshed. V.N. Pngodm, Solid State Commun. 50 (1984) 601.
0 0 3 8 - 1 0 9 8 / 8 5 $3.00 + 00 Pergamon Press Ltd.
TRAP CONTROLLED HOPPING IN ONE-DIMENSIONAL SYSTEMS V.N PrIgodln, Chr. Smdel* and A P Nakhmedov A.F. Ioffe Physlco-Technlcal I n m t u t , Academy of Sciences of the USSR, 194021 Leningrad, USSR
(Received 14August 1984 by A Zawadowsk 0 The results of a theoretical study of trap controlled hopping m 1D systems by means of the Berezlnsky techmque are presented. In the case of a discrete trap state we gwe exact time and field dependences of the relaxation current produced by rejected carriers. In an m~tlal time interval where the moving carriers are very effectively trapped one obtains exactly the result formerly found for the randomly broken bond model. Taking into account a small transverse hopping rate the 1D results are generahzed for quasl-lD systems The physical sense of the results Is d~scussed RECENTLY THERE IS high interest In the experimental study of orgamc materials (molecular crystals, polymers) being characterized by sufficient amsotropy and showing unusual transport properties [ 1 - 3 ] Various theoretical approaches have been used to mterprete the experimental data on the basis of hopping transport in disordered systems [ 4 - 8 ] Considerable success In the study of one-dlmensmnal (1 D) and quasi-1D systems, m partmular of thmr kinetic properties, has been achmved [ 6 - 8 ] using a special dmgram technique suggested by Berezlnsky [9] In the following we present the results of an exact study of the so-called trap controlled hopping m 1D systems and given a generahzatlon to the quasM D case The master equations governmg the charge carrier dynamms in the system under consideration reads
where w being an effective hopping rate without field and Vo is of the order of characteristic phonon frequencies u+ are u_ are capture and release rates of the trap whmh are assumed to be of the form u = 60exp (--et/kT),
u+ = co,
(3)
where e t is the trap depth and co is also of the order of characteristic phonon frequencies The total propagator Dnn'(t) IS given by
Dnn'(t) = Dnn'(t) + Dnn'(t)
(4)
with/)nn,(t ) = (fin(t)), Dnn'(t) = (Pn(t)) describing the transmon into a transport or trap state, respectively, at site n In Laplace representation
Dnn,(S ) = ; e-StDnn,(t)dt
(5)
0
dt
=
E (Wn,n+ fin+ --Wn÷ ,nfin)--U÷Pn
the current is determined by
g=+- 1
+u--P~
dL dt
](s) = eneas Z (n--n')Dnn'(s)
(6)
n
- u+fin--u--Pn,
(1)
where fin and *Pn are the conditional probablhtles to find at time t a moving carrier at site n in a "transport" state or in a trap state, respectively, if at t = 0 it was located in a "transport" state at site n', l.e, the boundary condition is fin(t = O) = 8nn' The transltmn rates wn, n+g depend explicitly on the apphed field E via the parameter 0 = eEa/2kT
wn+_qn - we = we +-°,
w = VOe-2a/rB
(2)
* Permanent address Instltut fur Polymerenchemle "Erich Correns", Akademle der Wlssenschaften der DDR, DDR - 1530 Teltow - Seehof, GDR 451
For solving (1) we use the Berezlnsky technique [9] The application of this method to the hopping problem in 1D systems has been described in detail [6, 10] Following the main ideas of this approach the interaction of moving carriers with randomly distributed traps is treated perturbatxonally The Green function Onn, is represented as hne from site n' to site n at which, similarly to the conventional cross technique, vertices are accomodated. Any diagram for Onn, can be decomposed into three blocks With respect to the external vertices at n and n' one has the left, the right and the central path Each of the blocks may be separately summed up Thus, consldermg blocks with a determined number of pairs of incoming and outgoing hnes, one obtains the following equations for the sum of
452
TRAP CONTROLLED HOPPING
right-part Rm(n ), left-part Lm,(n') and central-part diagrams K m,m(n', n )
Rm(n) = (z+/z-)-mnRm,
Lm,(n') = (z+/z_}m'n'Lm ,
Lm - Rm
(7)
Vol 53, No 5
scattering amplitude becomes f = -- 1, x e , the moving careers are trapped but there IS no release In agreement with (9) and (11), at f = -- 1 we have b Wmm, = (1 -- C)6mm, "t-C(-- 1)m~im,O
pVmm, = ( 1 - - C ) 6 m m , (z+lz_)mRm = [~ ~ Wmm'Rm'
(8)
m'
In this case we are able immediately to solve equations
(8), (10) and get
(.k~rn-k-lF2k+m'-rn ( .f)2 (m-/~) Wmm' = E ~ m ~ m ' - I J 1+ k vii
t
(9)
c ( - - 1) m Rm =
r
(z./z_) Km,m(n ,n) = D ~. Vmm"Km'm"(n ,n-- 1)
(z+/z-) m -- (1 -- c)
Km,m(n',r/) =
m"
[(1 --c)(z_/z+) m]n-n'-2(Z_/Z+)m~mm , at n > n'
at n > n'
Vmm'
(177
(10)
+f)2rn-2k+1
= E ~ ( 'm k t~~mk + ' m'-mj2k+m'-rn(1
k
(11)
Thus, equation (16) yields __
(~)m[( l - - c ) (z~)ml
~-'
~)nn,(S) = 1 f
n-n¢
zn_n,_
(12)
r = [(s + w÷ + w_) 2 - 4w_w÷]a'2
(13)
*
(z+ t
where
z._ = (s + w÷ * w_ +_r)/2w_
(18)
• (R,. + Rm÷l) Rm +~_gm.l
(191
Similarly we find
Together with the boundary conditions Ro = Lo =
1,
Km'm(n-- l,n)
= (Z_/Z +)mC~mm ,
L)nn'(S) = C E (-- l) mzn--n' (1 --C) $ m
(14) equations ( 7 ) - ( 1 1 ) form a complete set The quantity f i n (9) and (11) is the total scattering amplitude o f a moving carrier at a single trap
x (R m + Rm+l)
(20)
Equations (19), (20) are written for the case n > n', quite similar expressions are obtained at n < n' Hence the total current consists of two parts
H+/r f=
1 + u+/r + u_/s
(15)
The symbol/3 means averaging over all the possible trap configurations At n < n' one obtains slmdar expressmns for the central part. Combining the blocks and taking into account the various comblnauons with external vertices at n and n' one obtains
l(s) = ]free(s) +]tray(s)
(21)
with /tree(S) = 1°( 1 - - c ) 2 s
W
?"
u -6m -'(u 2 -
~m
1) 2 [1 - (1 - c ) : u - 4 m - 2 ]
[1 - - ( 1 --c)u-2m]2[1 - - ( 1 - - c ) 2 u - 2 m - 2 ] 2 *
z_n-n'(z+/z_) m÷ m'(R m, 4. Rm'+l) *
Dnn,(s ) = 1/r
I
rn, m t
• [e-°u -- (1 -- c)u-2m] 2 [e-°u -- (1 -- c)u-2m] 2 (22)
. K m , m ( n - - l n ) (, R m + ~ _ ) ( 1 6_) R' m + l . ]tr.p(s)
Similarly, the Green f u n c t i o n / ) . n ' can be written an terms of the three blocks As a first apphcatlon o f the basxs equations obtained above we discuss the case o f a discrete trap state occurring with concentratmn c, 1 e , in equations ( 8 ) - ( 1 1 ) with probablhty c we have u÷ 4 : 0 and with probablhty (1 -- c) u+ = 0 Following (15) in the highfrequency range given by the c o n d m o n u_/u÷ ~ s/r the
= 1o w
(1 -
c)c 2
E U-4rn+l(u2-- I)[1 - - ( 1 --c)2u -4m-2] m
[1--(1--c)u -2ml[1-(1-c)u
•
-2m-2]
1 * [ue_O
(l_c)u_2m ] [ueO_( l_c)u_2m]2,
(23)
where u 2 = z+/z_ and 1o = 2eneawshO is the current of a
Vol. 53, No. 5
TRAP CONTROLLED HOPPING
regular chain without disorder (traps). lfree corresponds to transmons where no trap state appears at site n, /trap to transmons into a trap state at n Summing up the two contnbutmns of the current one obtains precisely the expression known from the study of the randomly broken bond model [6, 7] This result is qmte surprising since we are concerned w~th two different physical problems described by different Green functions Their first moments, however, turn out to be equwalent. The same expression for the current is found if one starts from the representation
l(s) = 1o Y[ &,,(s)
(24)
?l
Equation (24) shows that only electrons occupying a "transport" state contribute to the current After carrying out the reverse Laplace transformation, in the weak field case (1 - c ) e ° < 1 one finds the following asymptotm time dependence at t ~ oo
l(t)/lo ~ exp [-- 2wt(ch ° -- 1)]
(25)
Since the current asymptotlcs contains no trap parameter, it is expected to occur due to field effects arising at the ends of clusters, I.e, m the nmghbourhood of traps. In the strong field case (1 --c)e ° > 1 large clusters gwe the mare contribution to the current asymptotms which is found to be 1
453
We note that the current asymptotlcs shows unusual behaviour for O/c -->0
j(t)/lo ~ exp { - - ( ~ )
'/3 [Tr(1--roal/2rar)/212/3}.(29)
In the field range 0 > c (c "~ 1,0 "~ 1) at t ~ ~o the current I (t) behaves as
\ ra, i - rd, ]
In the opposite frequency limit s/r ,~ u_/u+ the scattering amplitude f tends to zero. Then the sum in the equatlons for transfermatrlces Vmm,, W.,m, (9) and (11) can he restricted to terms where foccurs In the first order and one obtains Wmm'
= 8ram'
+ mf(6m',m+l +¢~m',m-1 + 28re'm) (31)
Vmm' = [1 + (2m + 1)f] 8ram' + (m + 1)f6m',m+l + mf6m,, m -*
(32)
It can be shown that expression (24) IS not related to a restricted frequency range but has general nature in the model under consideration [ 11]. Thus, we have
j(s) = ]oC,(q = O,s),
(33)
where G(q, s) is the Fourier transform of/3nn,(s )
[ ( 1 - - c ) 2 e 2 ° - - 112
c(1- - 3 t -o-F v - i]
j(t)//o
ex [-cw (o0
e-0
26,
The transition regime before reactung the asymptotms is more transparent m the hmit of low concentrations c "~ 1. For weak fields 0 ,~ 1 and 0 < e, one obtains
e-'q(n-n')Dnn,(s ) = O+(q)
+ D_(-- q) -- Doo with
Z+ Z-
D0o = 1/r E(g+lz-)m(Rm+Rm*l) R m + - - R m * l
(35)
1'2
+
I J \r~l
O±(q,s) = 1/r ~, Qm(q,s)(z+/z_) m
• exp --
(34)
*
[l_]_(~7"d-ifXel2]12t__~
r~
~
I'1= - ~
r¢l
2rrI/2xo
1(0/1o =
G(q, s) =
r ~ t - - 3 ( r r / 2 ) 2/3
X (1 --1 ;~)2 ~ ] 1/3}
m R=
Z+
+Rm+x - - , g-
(36)
(27)
-where D+ and D_ in (34) and (36) correspond to the cases n > n' or n < n', respectively In (36) we Introduced the auxiliary funcUon Q~(q, s) related to the original blocks by
where 1 ~'chf =
"'"
cZw '
1 "fdr --
2cw ° '
Xc
=
[
[ 1(
2t.__]rat~_ l ''3
Qm(q, ~ e-'qnQ~n( - 1,n) * s) = n=o
2 r~]]
Qm(q,s) = £ e-'onQm(-n, 1) .=o
(28)
(37)
454
TRAP CONTROLLED HOPPING
w~th
Vol. 53, No. 5
Finally for the current we get
Q~n(- 1, n)
=
1
z n_ [z+]m' I~_) (Rm'-t-Rrn'+l)
~,
× K'm'm(-- 1, n),
n > 0
_
Q~n(n, 1) = Zm, z+n [-~-) K,nm,(n, 1)(Rm, + Rm'+l)
(38)
n<0.
From (10), we obtain the following equation for a~n(q, s)
+ ~±(z+/z-) m * (Rm + Rm+~)
ch~o = 1 + 2u_/cu+
aim = ( - l)me-m~O(1 -- e - ~ Q o , 2u_ 1 Qo = cu+ sh~o (w/s)l/2
1
(39)
(40)
w± = Uo exp (-- 2/31r:); < w±
Due to the exit term m (44) there occurs a renormallzatlon of the bare Green funcUon Dnn, and of the parameters z±. For weak field, thin renormahzatlon gives effectwely an addmonal contribution to the field parameter 0, I e.,
(41)
The lncommmg transverse term is perturbatlonally treated Finally the Green function can be represented as
1 G(q, qi, s, 0") = G-:(q, s, 0") -- w±(q±)'
(50)
where G(q, s, 0") is the exact 1D Green funcUon (with corngated 0) and the Fourier transformed transverse hopping rate m
to the diffusion coefficient being equal to Do = a2w. Hence, the characteristic relation u_/cu+ of the trap parameters can be estimated by means of (42). At weak fields 0 < 1 in the equations for Rm, Q~ in (40), (41) instead of ~0there occurs the function ff given by
ch~ = ch~ + 2 U_w02/s
(44)
C U+
and in (41) Qo must be replaced by + 0
(45)
Q~ ~
wl(q: ) = w I ~ e'qXRu.
+0
(51)
In this way the current in the quam-lD system is found to be 1
101D/lo = 1 --ZWl(S, 0")/1o llD(s' 0")/Io
(52)
i.e., the asymptotic behavlour obtained above is maintained. As a result of the nonzero transverse hopping rate the field parameter 0 is changed only This effect can be understood as a change m the mean drift time
1/2"
C U+ S
(49)
W
(43)
-- --
(48)
(42)
/do = e/kTDo
sh~ ~ - -
(47)
where a l # are chain Indices The sum is extended over z± nearest-nelghbour chains The mtercham hopping rate w± is supposed to be much smaller than the lntercham ones
02 ~ 0 *2 = 02 + ziw----'~z
where Oo =/done Is related by the Einstein relation
2 u_ w CU+$
Conmdermg the cases Owls .~ 1,0 w/s >> 1 one obtains current asymptotlcs similar to (42) The results dmcussed so far can be generalized for a quam-1D system m which a small amount of mterchaln hopping is allowed In this case the master equation (1) must be completed by the term
I.t
I.e., at 0 = 0 we have Q~ = Q~n- The asymptotic conductwRy wtuch can be similar to (33) written in terms of G(q = 0, s), becomes
o(s)/oo = 1/s ~ , CU+ 1-t-u_
-- 1
s h ' - ~ (OF + Qo - 1) (46)
wz ~. Pu(s, u) -- ziwiPd(sl a),
with ~_+ = (z~) ~ 1 In agreement with (33), for calculating the current not the whole Q-function is reqmred but only Q~m(q = O) = Q~. Let us discuss several cases Without field, 1.e at 0 = 0, equaUons (8), (38) yield the following soluuon for R m, a ~
Rm = (-- 1)me ~m~°,
ch~
](s)/lo = 2w(s/w + 02) 1/2
Tdr
Tdr
T,.,r
T
TRAP CONTROLLED HOPPING
Vol. 53, No 5
with r± = w_~~. A similar result was estabhshed m the case of a continuous trap-depth distribution [12] In the field-free case the present result indicates different long-time behaviour of the diffusion constant for 1D and quasl-lD systems, respectwely.
5. 6. 7 8.
REFERENCES 1. 2.
3. 4.
D. Haarer & H. Mohrwald, Phys. Rev. Lett. 32 (1975) 1447. U. Selferheld, H. Bassler & B. Movaghar, Phys Rev Let. 51 (1983) 813 S. Klvelson, Phys. Rev Let. 46 (1981) 1344. J. Bernasconi, S. Alexander, W.R. Schneider & R. Orbach, Rev. Mod. Phys. 53 (1981) 175.
9. 10 11. 12.
455
G. Pfister & H. Scher,Adv. Phys. 27 (1978) 747. V.N. Prigodin & A.N. Samukhm, Preprmt A.F. Ioffe Physico-Techmcal Instltut Leningrad 1983823. V.N. Pngodm & A.N. Samukhm, Solid State Commun. 46 (1983) 379. V.N Prigodm, Zh. Eksp. Teor. Ftz. 86 (1983) 242. V.L Berezmsky,Zh. Eksp. Teor. Ftz. 65 (1973) 1251 V.N Pngodm, Zh. Eksp Teor. Fiz 82 (1982) 1221 V.N. Prigodin,Zh. Eksp Teor. Ftz. to be pubhshed. V.N. Pngodm, Solid State Commun. 50 (1984) 601.