- Email: [email protected]
Localized states versus band states in a model for small polarons
Physica 74 (1974) 559-576 0 North-Holland Publishing Co.
LOCALIZED
STATES VERSUS BAND STATES IN A MODEL FOR SMALL POLARONS 0. ENTIN-WOHLMAN
and D. J. BERGMAN
Department of Physics and Astronomy, Tel-Aviv University, Ramat Aviv, Tel-Aviv, Israel
Received 2 January 1974
synopsis A general formal theory of irreversible processes is applied to Holstein’s one-dimensional model for small polarons in order to find the appropriate representation of states for the system as a function of temperature. We find conditions under which the exact equations for the occupation probabilities of a band state or a localized state become markoffian. The temperature region in which a representation is adequate is determined by the markofficity conditions and by the requirement that it is possible to prepare the system in one of the states of that representation. We find that the band picture is appropriate at low temperatures while the local&d-states picture is adequate at high temperatures. Both representations become inadequate approximately at the same point.
1. Introduction. The motion of small polarons in ionic crystals is interesting because of its twofold character. At low temperatures the polaron moves by quantum-mechanical tunneling. Thus its states are suitably described in the band representation. The bandwidth is that of an electron in the tight-binding approximation reduced by a temperature-dependent exponential term which bears the influence of the strong electron-phonon interaction and which decreases with increasing temperature T1). Consequently, as T is raised, the bandwidth eventually becomes smaller than the uncertainty in the energy of a single band state. Then the band representation is no longer adequate and the appropriate description is in terms of localized states, among which the electron moves by phonon-activated hopping. In this article we present a method for finding a suitable representation for the states of the system as a function of temperature. Our method is based upon some general formal theories for irreversible processes, as presented by Zwanzig2) and Sewel13). The model we consider offers one of the few examples where these theories can be worked out in detail. Thus it throws light on some of the assumptions and approximations used in the theories. The method can also be applied to other systems. For example, one might consider the electron-phonon system 559
560
0. ENTIN-WOHLMAN
AND
D. J. BERGMAN
under the influence of an electric field and try to decide about the existence of a Stark ladder. The method may even be useful in the discussion of the intermediate region between localized and nonlocalized states. We will begin our discussion by assuming a certain representation for the states (i.e., either localized or band states). We then derive an exact linear integrodifferential equation for the time-dependent occupation probability P,,(t) of the nth state. The equation cannot be solved exactly even in the model. We then search for conditions under which the equation becomes approximately markoffian. The approximate markoffian equation then enables us to calculate the probabilities at any time t from a knowledge of the probabilities at an earlier time, if these times are separated by more than a characteristic microscopic time. This microscopic time is the characteristic lifetime of the microscopic processes which cause transitions among the states. Thus the markofficity implies that if a state of the system is described by a linear combination of the states of the representation, then the phase relations among the coefficients are not important for the description of the temporal evolution of the system. In this sense the states of the representation form a good approximation to the eigenstates of the system. The solution of the markoffian equation is subjected to the condition that the system can be prepared in one of the states. This solution determines a temperature-dependent lifetime of a state and consequently gives the temperature regime in which the a priori assumed representation is adequate. In section 2 we calculate the occupation probability of a localized state and the transition probability per unit time for a hopping process. In section 3 we derive the occupation probability of a band state and calculate the lifetime of this state. The localized-states picture is adequate at high temperatures while the band picture is suitable at low temperatures. Both representations become inadequate approximately at the same temperature, which is the transition temperature between the regions of hopping and tunneling motion. It has been shown elsewhere4) that the resulting transition temperature is in agreement with experiments. Our treatment of the small polaron motion is based upon Holstein’s’) model. The model consists of an electron moving in a one-dimensional molecular crystal made of N diatomic molecules of mass M. It is assumed that the centre of mass of each molecule is fixed and the motion of the electron is treated in the tightbinding approximation. Holstein’s’) hamiltonian can be written in second quantization in the form: H = HP,, +
xCho,(A;b, ”
4
+ A”_,b;)C,tC,, -
J; (C$2n+, +c,tc,_,).
(1)
n=1
Here : f&h = c Am, (b;b, + :) 4
(2)
LOCALIZED STATES VERSUS BAND STATES FOR POLARONS
is the hamiltonian of the lattice vibrations in the harmonic approximation,
w;+ co;cos qa
2
CDIl=
561 where: (3)
and a is the lattice constant, AI = - A elana (1 /~Au$‘F~o,“)~,
(4)
where A is the strength of the electron-phonon interaction. Cf; (resp. C,) creates (resp. annihilates) an electron localized at the nth site and J is the overlap integral governing the transfer of the electron between nearest-neighbour sites. In ionic crystals (like NiO, MnO, COO) the number of electrons is small compared with the number of lattice sites, so that the electrons move approximately independently of each other. Therefore, considering the motion of a single electron, Holstein assumes that the wavefunctions of the system can be written in the form: x. aAN,) Cf; IO>, wh ere a,(N,) is the wavefunction of the lattice vibrations when the electron is localized at the nth site, and N, are the occupation numbers of the phonon states. We proceed somewhat differently: by applying’ a unitary transformation T = exp
-c (
nil
Ai (b, - bL4) CJCn , )
(5)
we first transform the hamiltonian to a form more convenient for our calculations. Noting that: = exp
TCiT’
(
-c
4
Ai (b, - b!_,,) Ci, )
Tb,Ti
= b, - 1 A”-&% n
(6)
we obtain: THTt
= Ho + V,
(7)
Ho = f&h - cc AcogA:A”_,C,‘Cn, ” Q
(8)
v = 1 c Lemnfc,
(9)
”
e=*1
Vnn+e = -Jexp
-c (
4
(A: - A:+e) (b, - bt,& . )
H,, , which arose from applying the transformation
(10)
to the first two terms of eq. (l), now has the simplifying property that the electron and the phonon variables are completely separated, whereas they were completely mixed before the trans-
0. ENTIN-WOHLMAN
562
AND
D. J. BERGMAN
formation. Consequently, the one-electron eigenfunctions of H,, are
where 10) is the electron-phonon vacuum state and 19is an arbitrary function of the phonon creation operators. To obtain the form of these functions in the original representation, we apply the inverse transformation T-l = Tt to (lOa), noting that Tt IO) = IO): T+C,t0 (b;) IO) = T+C,tTT+e (b;) TT+ IO)
The last result justifies the assumption made by Holstein and allows us to identify
dNQ)
a&V,)
=
exp 1 Ai (b, - bt_,) 0(6:) 10). 1 (Q
It also permits us to identify the operator which creates a single dressed electron (i.e., a polaron) at the site n as TtCJT
= exp x Ai (b, - b!_,) CJ. > (4
The second term in H,, represents the binding energy of the polaron. This is the energy needed to localize the electron near the site n and to displace the internal elastic coordinates of the molecules at and around that site. The effect of the unitary transformation Ton H is to cancel the usual electron-phonon interaction [Le., the second term in (l)]. On the other hand. the overlap integrals J acquire a phonon dependence. 2. The occupation probability of a localized state. We now derive an equation of motion for the probability that a localized state is occupied at a time t. The derivation is based upon the formal theories of irreversible processes developed by Zwanzig’) and Sewel13). In these theories averages are defined with respect to the microcanonical ensemble, and consequently the application of such theories to the electron-phonon system is cumbersome. We shall define averages with respect to the canonical ensemble. This modification is not trivial and causes alterations in the basic formalism of the above-mentioned theories. The Hilbert space of the system is spanned by the wavefunctions of H,, [eq. (S)]. We divide this space into N cells according to the eigenvalues of the number
LOCALIZED
STATES VERSUS BAND STATES FOR POLARONS
563
operators CiC,. In this way each cell characterizes a situation in which the electron is localized at a certain site and the phonons are free to be in any of the states e(bb) 10). The part Y [eq. (9)] of the hamiltonian causes transitions among these cells. We define a coarse-grained projection operator 9 which acts upon operators in Hilbert space : 90 = c Tr (SC,tC,,j (l/W,) e-B”pbC,?C,, ”
(11)
W,, = Tr (e -BHphC.tC,).
(12)
When B operates on any operator 0, it replaces it in every cell by the diagonal operator e -BHphCtC n “, suitably normalized so that the trace in that cell is the same as for 0. BO is called the gross part of 8 since it coarse-grains over the details of the microscopic states in each cell. The probability P,(t) that the nth localized state is occupied at a time t is:
p,(t) = Tr (e(t) CJG),
(13)
where e(t) is the density matrix of the system. The evolution in time of P,,(t) is determined by the evolution in time of e(t). To find the latter, we introduce the Liouville operator 8: 9==!?0+8,,
(14)
900 = No, 01,
(14a)
910
(14b)
= [V, U]
and obtain
e(t)= @l(t)+ xe
@z(O)- (i/A) i dt’ (1
-wti)recl-s)t’
0
2
(&
(t
_
where @l(t) = pe (0,
(153)
e&j = (1 - 9) e(t).
(15bj
We assume that at the initial time t = 0: e(O) = C e. Cl/ WJ ewBHphCC,, n
564
0. ENTIN-WOHLMAN
AND D. J. BERGMAN
where the numerical coefficients en give the initial weight of each cell. If en = h,;; then at t = 0 the electron is localized at the llth site, while the phonons are in thermal equilibrium. This seems to be a realistic initial condition, because (as argued in the following) the lifetime of the electron in a localized state is much longer than the characteristic time of the phonon processes. The initial condition implies that: 9~ (0) = ~~(0) = e(O), so ~~(0) = 0. Inserting this into (15) and using (13) we obtain :
dpn (t) = i ’ Tr [C~C,$‘~ (f)] = -$- 1 i dt’ n’ 0
dt [(CJC,IC,)
Tr
x
e-(i,‘h)u(l-s)t’
x (l/W,,> P,,, (t
$p
(,-%h
C$-+)]
- t’).
(16)
To get this result we also used the relations
which can be verified from (11) and (14). Since we have not made any approximations
in deriving
equation for P,(t). In order to solve it, we shall now investigate the kernel. We write this kernel in the form: (I//PW,.) I(O)(f)
=
nn’
~$(t’)
Tr [9 (CiC,J e-(i’A)r(l-@)t’
-l-r
[(y
C 1
= (-i/h)’
n+
C
n
)
e-“‘“‘zO”
g2cr)
=
e-‘i”‘“O”
2,
of
(17)
(17a)
(tl) ..* _Y2(fI)
(1
_
uj3)
I#
0,
e-ci~fi)oOre
1 ((Vn’n’+E(f’) V”,,,,.) &=+l
(17b) (17c)
We now calculate the first few orders of the kernel The zeroth order, from (8)-(IO), (14) and (17a) is:
r;;?(t’) = (l/P)
= t I$ (t’), 1=0
(ewBHphCJL$,)] (I/ W,A”),
9,
(ePBeph CJCn,)] (l/W,A’),
eWfi).LPOf 6?p,
it is an exact
the properties
11-l ...j dt,
j’dt,
x Tr [(~,C~C,,) x 2,
B (eePHph C$C,,)]
(16)
and discuss the general
+ cc.) (&I”*+, - &“,).
term.
(18)
LOCALIZED STATES VERSUS BAND STATES FOR POLARONS
565
Here (O,,), where ODhis a function of phonon operators, is the equilibrium and O,,(t) = e”‘fi’HDh’ OPhe-(l’fi)H@‘r.In order average : Tr (e -BHnhO,,J/Tr (eWBBph) to calculate higher orders we note that:
~&> 0 = VW, 01 - c O/K,) Tr (OC~IG,)MO, ewBHph C.‘,G,l. “1
(19)
The first-order term is: 1’l! nn (t’ 3 tl) = -(i/A” W II’) Tr (e -BHDhCX”,
[K WI),
[W’),
CGlll>.
Explicit calculation of the commutators yields that this expression consists of 8 correlation functions of VnnLt 1 of the type : E BC+l(V~,“,+rVn’+~“‘+d+E’ 01) v”,+e+el+“4’))4+,,,*1~““.. 9,
-
Since E + &I can be either zero or +2 these correlation functions vanish. It is easy to see that, in general, any odd-order term of the kernel includes a delta function which makes the whole expression vanish unless the sum of an even number of E , each of which is 5- 1, is equal to + 1. Since that is impossible all the odd-order terms of the kernel vanish. The second-order term is :
1;;) (t’tlt*)
= -(l/PWn,)
x Tr (ewBHDh C$G W, [J%), [WA VW>, C,!C,llll> +-
l PW,.
CL
Tr (emcHphC,$C,,, [V, [ V(t,), C,?lCn,]])
Ill w,,
x Tr (eeBHph Cn’,G,PW, WV), C!C.ll> and consists of 16 correlations of V4 minus 16 products of two V2 correlations. A typical V4 correlation is:
0. ENTIN-WOHLMAN
566
AND
D. J. BERGMAN
where we have defined: ~“‘“2 E A;’ - A;’ 4
(20a)
and K.
E
(( Vn,+,))2 = J2 exp
A2 -- NMfi
1 - cos q” coth t?!!. c4 UJ3 2 g
.
(21)
>
Kt decreases exponentially when T increases and is always smaller than J’). We need to know the behaviour of the correlations at long times. This behaviour is found by the method of steepest descent which gives:
x cos
[(cog- co:)”(tl - ti + C0, (fl - I,)+
$$A) 4 +7r]
(22)
Thus, if any of the time differences in (20) is much larger than I/C!), , the suitable sum in the exponent tends to zero. This decay property entails that whenever the difference between the times (t’, ti) and (f2, 0) is larger than l/to, the V4 correlation is decoupled into a product of two V2 correlations. This does not imply, however, that I,$) (t’, tl, t2) tends to zero. We have listed the diagrams which remain in Zii? (t’, tl , t2) in fig. 1. The correlations that are described by these diagrams tend to Kg when all the time differences in (20) are larger than l/~,j,. It is clear that these diagrams would appear even if one were to calculate the second-order term in a system without any electron-phonon interaction. In such a system A = 0, T = 1 and Vnn+C = -.Z, so that the second-order terms would be given by the diagrams of fig. 1, where each diagram is now multiplied by J4 instead of Kz. It will be shown in the next section that 4K$ is indeed the bandwidth of the polarons. The diagrams which remain when all the time differences are much larger than l/w, describe the virtual-phonon processes which manifest themselves in the temperature dependence of the bandwidth. The calculation of Z,‘f!(t’, t, , . . . , tL) for an even I > 2, is quite straightforward, although cumbersome. The Ith order consists of 2’+’ correlations of order V+ 2, minus all their decouplings into products of V2, V4, . . . , V’ correlations, all of which involve sequential times. But again, even when the longest time t’ is much greater than I/o,, the Ith order does not vanish. This failure of the kernel to vanish even for times much greater than l/o, which appears in every order, is worthy of note. In the general theory developed by Sewel13) it was assumed that
LOCALIZED
STATES VERSUS
BAND STATES FOR POLARONS
567
if after a rather short time (Le., short compared to the lifetime of a state) the correlations decouple into products of lower-order correlations, the kernel vanishes. In the next section we will see that this vanishing property does hold for the bandstates representation.
VW
”
n’-4anI+, Ws)
v(t,)
VU.)
Y
n8-f~nB+4
n”-t~n~+9 VW )
nbmn1+2
VW)
&2-n* 4
VW)
602)
v(t,
1 If1 v(t’)
v(t, 1
n’c2Qno+2 ”
2
Fig. 1. The diagrams which describe the correlations that remain in I,$f? (t’, t,, tz) when I’, tt and t2, 0 are separated by an interval % l/o1 . The vertices indicate lattice sites and an arrow accompanied by V(t) between two vertices n, n + E represents the operator V,,,+,(t) C.‘C.+.. The correlation described by the first diagram is: < V,,,,,,_i I’,,._,,, (ti) I&,+i(t’) &f+ln,(tz)), and it appears three times in I,$f? (t’, tl , tJ because in this diagram n can be equal to n’, n’ k 1.
From eqs. (20) and (22) it can be seen that the second-order term is negligible compared to the zeroth order if the following conditions hold: J/fic0, < 1,
(23)
K$‘/k < 1.
(24)
The reason for this is as follows: any sum over q in the exponent of eq. (20) differs from zero as long as the time difference appearing in it is B 1I@,. We approximate the sum in this case by its maximum value, attained when the time difference in zero. Thus, that part of the exponent together with K$ gives J. When the time difference is > l/o,, the sum tends to zero [eq. (22)]. Therefore, in this region the exponential is approximated by 1. The same arguments hold for the I > 2 orders of the kernel. Consequently, when (23) and (24) hold, all the 1 > 0 orders
568
0. ENTIN-WOHLMAN
AND D. J. BERGMAN
of the kernel are negligible compared to the zero order. The physical meaning of these conditions will be discussed later. Under these conditions the equation for P,(t) [from eqs. (18), (20) and (21)] is:
dpn Ko = $y egkI(Bnn.+, - b,,.) dt
f dt’P,,s (t -
A2 (1 -cos
qu) cos ~0~(t’ + _ti@)
NM&o,3 We now write the term in the brackets
(
exp C [
A2 ;;;;;,a)
t’)
0
sinh &%%x,
..1.
+ cc >
on the r.h.s. in the form:
cos;;;;;
+ C.C. - 21 + 2
Q
and divide it into two parts. The second part would have remained even if there were no electron-phonon interaction (multiplied by J2 instead of Ko). It thus bears the influence of the virtual-phonon processes, and consequently contributes to changes of P,(t) due to tunnelling. The first part vanishes for w,t’ 8 1 [see eq. (22)] and describes phonon-activated processes. If we consider P,(t) at times that satisfy:
the contribution
of the second
term to dP,,ldt becomes
markoffian,
and we ob-
tain
dp,,
-
dt
=,=j,
-
(4m,+,-
S,,,)
K,P,,
2Ko
(t) + -
82
(26)
where (27) -CO
It is easy to see [by changing the variables in eq. (26): K,t = t; P,(t) = p,,(7)] that the relative importance of the two terms on the r.h.s. of eq. (26) is determined by the factor K$X, . It is shown in the following that Kl is the transition probability per unit time for a hopping process. Thus, Ki/fiK, is the ratio of the transition probability of a tunneling process to that of a hopping process. K. is an exponentially decreasing function of T which attains its maximum value at T = 0 [see eq. (21)], while Kl vanishes at T = 0 and increases as the temperature is raised:
KoU’J13/W (Td > CKo(G)13/%(TA
T, < T2.
LOCALIZED STATES VERSUS BAND STATES FOR POLARONS
569
Therefore, at sufficiently high temperatures which satisfy
the hopping mechanism dominates over the tunneling mechanism. That is, the second term on the r.h.s. of eq. (26) is negligible compared to the first and consequently, P,(t) N eezKlt 1,-1 (2K,t),
(29)
where I,,(X) is a modified Bessel function. Here we assumed that P. (t = 0) = a,, 1. [To solve eq. (26) when the second term is negligible we Fourier-transformed it, solved for the Fourier transform, and then found the inverse Fourier transform of that solution.] From (29) we see that in the temperature region in which (28) holds, K;l is the characteristic lifetime of a localized state, and Kl is the transition probability per unit time for a hopping process. The transition probability increases with T because the number of accessible phonons increases with T and thus the phonon-activated processes become more frequent. At temperatures higher than the Debye temperature, Kl assumes the form of a thermal-activated transition probability’*4). We now examine the conditions under which the solution (29) was obtained. The first condition, eq. (23) is temperature independent and determined by the physical properties of the system. It means that the tunneling frequency J/hof the bare electron must be smaller than the characteristic frequency of phonon processes. The second condition, eq. (24), reduces to eq. (28) when t’ is replaced by the lifetime K;l. Thus, a consistent solution is obtained in the range of temperatures determined by (28), in which the hopping mechanism dominates over the tunnelling mechanism. When eq. (28) holds, the second term on the r.h.s. of (26) is negligible compared to the first term, and eq. (26) becomes markoffian. Since in order to obtain (26) we used eq. (25) co,/K,% 1 must hold for a consistent solution. Namely, the lifetime of a state must be much longer than the microscopic time l/w, which is the lifetime of the integrand in Kl [eq. (27)]. Thus, both conditions w,/K,% 1 and K$/AK,4 1 are needed for the markofficity of the equation for P,(t), and they determine the temperature region in which the representation by localized states is adequate. In the temperature region in which the tunneling mechanism dominates over the hopping mechanism [i.e., when eq. (28) is violated] the first term on the r.h.s. of eq. (26) is negligible compared to the second, and the characteristic time of P,,(t) is of the order A/K$. Going back to (24) we see that the I > 0 orders of the kernel are not negligible compared to the zero order. If we neglect the realphonon processes, that cause the hopping, in the kernel and keep only the virtual. phonon processes, the order of magnitude of the Ith-order terms of the kerne [eq. (17)] is (K,,/A') (Kt f/h)'. Thus, the characteristic time of the kernel is the
570
0. ENTIN-WOHLMAN
AND D. J. BERGMAN
same as the characteristic time of P,(t), i.e., both are equal to the tunneling time A/K;. Consequently, the equation for P,,(t) is not markoffian. Note that solving for P,(t) in this case is equivalent to solving the equation in the absence of the electron-phonon interaction. We show in the next section that in the region of low temperatures where Klf/#iK, > 1 the appropriate representation is by band states. 3. The occupation probability of a band state. In order to derive an equation for the evolution in time of the occupation probability Pk(t) of a band state k, we must construct the phase cells in such a way that each cell represents a state in which the electron is in a state k, while the phonons are in any of the states 8(bi) IO). We first Fourier-transform the hamiltonian (7) to obtain: H=H,+H,,
(30)
H, = HPh + c (ek - Eb) C,tC,,
(31)
k
&k
=
2K$ cos ka,
(314
(31b)
E,, = 1 hq,A;A)tq,
H alk2 = (l/N)
c n
c
(Vnnfe - K,f) e-iol-kZ)nufi’@,
(324
&=kl
where we have defined: C,t = (l/N”) T eeikna C,t
and k lies in the first Brillouin zone of the one-dimensional lattice. The vector space of the system is now spanned by the eigenfunctions of H,, and it is divided into coarse-grained cells according to the eigenvalues of the number operators c,tCk. Thus, C,tck is the projection operator on the kth cell. The part H, of the hamiltonian does not cause transitions among the various cells. It follows from (31a) that 4K$ is the bandwidth of the electrons when the electron-phonon interaction is taken into account, i.e., it is the bandwidth of the dressed electrons* l*“). The probability that a band state k is occupied is Pk(t)
=
Tr
[e(t)
c,tckl.
* In ref. 5 the polaron bandwidth in a three-dimensional lattice was calculated iational method. Kt is the one-dimensional version of that result.
(33) by the var-
LOCALIZED STATES VERSUS BAND STATES FOR POLARONS
571
Running through the same calculations as in the last section we obtain the same equations as (16) and (17) except that H,,, V, Cf;, C, are now replaced by K, Hi, CL Ck. The calculation of the orders of the kernel in the equation for dPk (t)/dt is carried out as in the previous section. In the present case, correlations of an odd number of Hklkl do not vanish. The Ith-order consists of 21+2 correlations of (I + 2) Hklkz, minus all their decouplings into products of Htlk2, H&, .*. Hk,,+ correlations, where all the correlations involve sequential times and each HklkZ(t) introduces a factor e(i’fi)(ek~-ek~)fdue to the time dependence of Cl,C,,. The important difference between the kernel in the equation for dP,/dt and the kernel in the equation for dP,/dt is found when t’ > l/o,. For these times all the orders of the present kernel tend to zero. We exemplify this behaviour by considering the second order of the kernel. It consists of 16 terms of the form:
(fbklHklk2 (t2)H&d g
exp
{(i/h)
[t&k1
02))
-
&k’)
H/&t’)) t2
+
c&k’
H,,,(f)))
-
&k)
ttl
-
t’)l)
*
When t2, tl < l/w, the time t’ is separated from all the other times including the time 0 and consequently [see eq. (22)] H&&t’) becomes uncorrelated with the other P&k,. Since (H,&> = 0 [from eq. (32a)] the two terms in eq. (34) vanish. Similarly, when t’, t2, t, > l/m,, the time 0 is separated from all the other times and again (34) vanishes. When t’, tl ‘> l/o, and 0, t2 < l/w,, the first correlation is decoupled and becomes: (Hk’kl HklkZ(tz)> 1 eq. (34) tends to zero. It is quite straightforward to check that similar cancellations occur in every order of the kernel. It is possible to prepare the system in a certain band state only as long as the minimum temporal width of a wave packet, tZ/K$, is not larger than the lifetime of a state, i.e., as long as: (&/fi) T > 13
(35)
where t is the lifetime of a band state. If this condition is violated, then the uncertainty in the energy of a single band state will exceed the bandwidth and the band picture is certainly inadequate. From (35) and the fact that J < ho, [see
572
0. ENTIN-WOHLMAN
eq. (23)] and K$ < J it already olz
> Jr/A > Kt+i
i.e., the condition
AND D. J. BERGMAN
follows
that
> 1,
(36)
for markofficity
is satisfied.
As a result of this, the exact equa-
tion for dP,ldt, y
= $dt’;
Pk> (t - t') GkkQ’),
[where Gkkf(t’) represents the kernel (17) when H,, HI, Ci, Ck, respectively] can be approximated tion = ; P,,(t) j%G,,,
y
H,, V, CA, C, are replaced by by a markoffian master equa-
(f’).
The upper limit of the integral was taken to co because of the decay property of the kernel. The Zth order of the kernel is of the order (K,/Fz’) (J/ho,)‘, thus from eq. (23) the I # 0 orders are negligible compared to the zeroth order. Note, however, that it is not necessary to have J < ko, in order to obtain markofficity in the band representation. If J 3 fro, the calculation of the kernel merely becomes more complicated as all orders then have to be taken into account. Keeping only the zeroth order, the master equation is
dP, (0 = _
eBp(W -&*) Pk,(t) -
5 [akk,
mkjk
eiB (i’n-‘x’) Pk(t)].
(37)
dt where
cc
(i/A) (t&s
x jdte -
x
Cl
-.Qt
CL
_
emla)
(1
_
e-igea
‘OS @I+ sinh $3ri~, >
The solution
of eq. (37) can be written
k’#k
1
1 .
(374
in the form:
Pk(t) = emxkf P, (0) + J’dt’ 1 AkkpPkT (t - t’) eexkr’, 0
_
(38)
LOCALIZED
STATES VERSUS BAND STATES FOR POLARONS
573
where xii = c Aktk,
(384
k’itk
A kTk = e - fS(W -Ed
&k’k*
Wb)
In order to find the lifetime of a band state k we rewrite the differential equation for Pk(t) in the following form dP, dt
= -XkPk + c &‘Pk’.
(38c)
k’fk
From (37a) it can be shown that all the A kk’ that we need are of the same order of magnitude (in fact, all of them are w l/N), and thus that
Since c Pk = 1, the second term on the r.h.s. of (3%~)is like an average value of Akk,, and is therefore also of the order 0 (l/N) compared to xk. Consequently, so long as Pk = O(l), we can neglect the second term and write dP,/dt % -XkPk. Thus the decay time of Pk is l/xk. In order to evaluate xk we first write akkJ in the form Olkk,=
(~O/h2)
;
eWfi)(e,~
-%)t
(Z/N)
1
--oo
eUk-k’ha
n
x [(eLn(‘) _
1) +
(esLnct) - l)cos(k
Na L,(t)
=
4AZ2 _!!$ ~0s
Mfi
nqa
sin2
x s ma
25
+ k’)a],
‘OScogt
2 sinh $3tZw,’
(39) (394
0
By inserting (39) into (38a) we obtain: xk
=
(2~,/h2)
j
dt
e-(*/h)ek(t+fiSh)
-cc
x C
”
eikna(i)” (eL’.“’ - 1) J, (z$(,+!E))
+ + (e-W)
I _ l)[e+V.,,
- eiko J ”_ l(2+yt+y))]}.
(?(I
+ T))
(40)
574
0. ENTIN-WOHLMAN
AND D. J. BERGMAN
We change t + $i/?fi -+ t and make use of the property L,, (t + $/?A) -+ 0 as o,t B 1 (this can be seen by the method of steepest descent): ,yk
=
(2&/fi2)
7 dt
e-ci’kr/fi)1 eikna(i)”
Because for o,t > 1 the integrand tends to zero, the important contribution to the integral comes from the times w, t d I. From (23) we see that in this region the arguments of the Bessel functions are very small : (Ko)/fi) t < (J/h) t < co,t < 1 and e -i(Eht/fi) tends to 1. Therefore xk c (2K,/ti2) By averaging cxk)
=
i dt [eLoCt)--co
this expression
we can write xk as follows: 1 + sin ka (edL1”’ -
I)].
over k we find (41)
2K,,
where K, was defined in eq. (27). Therefore, the mean lifetime in a band state is of the order KL’. The mean lifetime in a band state increases as the temperature decreases, and at T = 0 it tends to infinity. I.e., at zero temperature the band states become the eigenstates, as expected. Inserting z z KY’ into eq. (35) we see that the band representation is appropriate at low temperatures. From eqs. (28) (35) and (41) it now seems that at the temperatures
which satisfy
both the band picture and the localized-states the adequate representation would probably over a finite number of lattice sites.
picture are unsuitable. In this region be in terms of states which are spread
4. Discussion. We have presented a method for finding the suitable representation of states as a function of the temperature for a model system with strong electron-phonon interaction, As this model forms one of the few examples in which the formal theories of irreversible processes can be worked out, it enabled
LOCALIZED STATES VERSUS BAND STATES FOR POLARONS
575
us to examine some of the assumptions and approximations which are made in these theories. In section 2 we presented a modified derivation of the equation for the occupation probabilities, in which averages are calculated using the canonical ensemble. In the case of localized states we have found that although the two products V(t,) .=a V(t,) V(t’) and V(t,+,) ... V(t,) V become mutually uncorrelated whenever their respective temporal ranges (t,, t’) and (0, t,, 1) become separated by more than l/w,, the equation for the probability does not become markoffian. This possibility was overlooked in ref. 3. Eq. (16) for the probability J’,(t) and the analogous equation for Pk(t) were derived exactly, without any coarse-grained approximations. It is only when one tries to calculate averages using P,(t) that one makes coarse-graining approximations. That is, one approximates the average of any operator 0 = Tr [e(t)
01,
by using the coarse-grained part PI(t) = SQ (t) in place of e(t):
where 0, is the equilibrium average of 0 in the nth cell: 0, = (l/ W,,) Tr (emDHph OC,?C,). Since it is impossible to solve the exact equations for the occupation probabilities, we found markofficity conditions for each equation. We then solved the approximate markoffian equation to obtain the temperature-dependent lifetime of a state and to find the temperature region in which the representation is adequate. In the region near the transition temperature [given by eq. (42)], neither representation is adequate. Therefore in this region neither the band states nor the localized states form a good approximation to the true eigenstates of the system. The problem of the proper intermediate representation can also be solved by our method. Assuming a certain representation for the states in the intermediate region, the phase space should be divided into cells according to these states. Then the exact equation for the temporal evolution of the occupation probability can be found. The decay properties of the kernel in this equation will yield the markofficity conditions for the equation. If the markoffian equation can be solved to give a lifetime which is consistent with these conditions, and if it is possible to prepare the system in the proposed states, then the representation is an adequate one. Acknowledgements. many helpful discussions.
One of us (O.E.W.) would like to thank P. Gluck for
576
0. ENTIN-WOHLMAN
AND D. J. BERGMAN
REFERENCES 1) 2) 3) 4)
Holstein, T., Ann. of Phys. 8 (1959) 343. Zwanzig, R.W., Boulder Lectures in Theoretical Physics (1960). Sewell, G.L., Physica 34 (1967) 493. Entin-Wohlman, O., Ph.D. thesis, unpublished (1972). Clark, P.M.. Gluck, P. and Wohlman-Entin, O., J. of stat. Phys. 7 (1973) 361 5) Sewell, G.L., Phil. Mag. 36 (1958) 1361.
LOCALIZED
STATES VERSUS BAND STATES IN A MODEL FOR SMALL POLARONS 0. ENTIN-WOHLMAN
and D. J. BERGMAN
Department of Physics and Astronomy, Tel-Aviv University, Ramat Aviv, Tel-Aviv, Israel
Received 2 January 1974
synopsis A general formal theory of irreversible processes is applied to Holstein’s one-dimensional model for small polarons in order to find the appropriate representation of states for the system as a function of temperature. We find conditions under which the exact equations for the occupation probabilities of a band state or a localized state become markoffian. The temperature region in which a representation is adequate is determined by the markofficity conditions and by the requirement that it is possible to prepare the system in one of the states of that representation. We find that the band picture is appropriate at low temperatures while the local&d-states picture is adequate at high temperatures. Both representations become inadequate approximately at the same point.
1. Introduction. The motion of small polarons in ionic crystals is interesting because of its twofold character. At low temperatures the polaron moves by quantum-mechanical tunneling. Thus its states are suitably described in the band representation. The bandwidth is that of an electron in the tight-binding approximation reduced by a temperature-dependent exponential term which bears the influence of the strong electron-phonon interaction and which decreases with increasing temperature T1). Consequently, as T is raised, the bandwidth eventually becomes smaller than the uncertainty in the energy of a single band state. Then the band representation is no longer adequate and the appropriate description is in terms of localized states, among which the electron moves by phonon-activated hopping. In this article we present a method for finding a suitable representation for the states of the system as a function of temperature. Our method is based upon some general formal theories for irreversible processes, as presented by Zwanzig2) and Sewel13). The model we consider offers one of the few examples where these theories can be worked out in detail. Thus it throws light on some of the assumptions and approximations used in the theories. The method can also be applied to other systems. For example, one might consider the electron-phonon system 559
560
0. ENTIN-WOHLMAN
AND
D. J. BERGMAN
under the influence of an electric field and try to decide about the existence of a Stark ladder. The method may even be useful in the discussion of the intermediate region between localized and nonlocalized states. We will begin our discussion by assuming a certain representation for the states (i.e., either localized or band states). We then derive an exact linear integrodifferential equation for the time-dependent occupation probability P,,(t) of the nth state. The equation cannot be solved exactly even in the model. We then search for conditions under which the equation becomes approximately markoffian. The approximate markoffian equation then enables us to calculate the probabilities at any time t from a knowledge of the probabilities at an earlier time, if these times are separated by more than a characteristic microscopic time. This microscopic time is the characteristic lifetime of the microscopic processes which cause transitions among the states. Thus the markofficity implies that if a state of the system is described by a linear combination of the states of the representation, then the phase relations among the coefficients are not important for the description of the temporal evolution of the system. In this sense the states of the representation form a good approximation to the eigenstates of the system. The solution of the markoffian equation is subjected to the condition that the system can be prepared in one of the states. This solution determines a temperature-dependent lifetime of a state and consequently gives the temperature regime in which the a priori assumed representation is adequate. In section 2 we calculate the occupation probability of a localized state and the transition probability per unit time for a hopping process. In section 3 we derive the occupation probability of a band state and calculate the lifetime of this state. The localized-states picture is adequate at high temperatures while the band picture is suitable at low temperatures. Both representations become inadequate approximately at the same temperature, which is the transition temperature between the regions of hopping and tunneling motion. It has been shown elsewhere4) that the resulting transition temperature is in agreement with experiments. Our treatment of the small polaron motion is based upon Holstein’s’) model. The model consists of an electron moving in a one-dimensional molecular crystal made of N diatomic molecules of mass M. It is assumed that the centre of mass of each molecule is fixed and the motion of the electron is treated in the tightbinding approximation. Holstein’s’) hamiltonian can be written in second quantization in the form: H = HP,, +
xCho,(A;b, ”
4
+ A”_,b;)C,tC,, -
J; (C$2n+, +c,tc,_,).
(1)
n=1
Here : f&h = c Am, (b;b, + :) 4
(2)
LOCALIZED STATES VERSUS BAND STATES FOR POLARONS
is the hamiltonian of the lattice vibrations in the harmonic approximation,
w;+ co;cos qa
2
CDIl=
561 where: (3)
and a is the lattice constant, AI = - A elana (1 /~Au$‘F~o,“)~,
(4)
where A is the strength of the electron-phonon interaction. Cf; (resp. C,) creates (resp. annihilates) an electron localized at the nth site and J is the overlap integral governing the transfer of the electron between nearest-neighbour sites. In ionic crystals (like NiO, MnO, COO) the number of electrons is small compared with the number of lattice sites, so that the electrons move approximately independently of each other. Therefore, considering the motion of a single electron, Holstein assumes that the wavefunctions of the system can be written in the form: x. aAN,) Cf; IO>, wh ere a,(N,) is the wavefunction of the lattice vibrations when the electron is localized at the nth site, and N, are the occupation numbers of the phonon states. We proceed somewhat differently: by applying’ a unitary transformation T = exp
-c (
nil
Ai (b, - bL4) CJCn , )
(5)
we first transform the hamiltonian to a form more convenient for our calculations. Noting that: = exp
TCiT’
(
-c
4
Ai (b, - b!_,,) Ci, )
Tb,Ti
= b, - 1 A”-&% n
(6)
we obtain: THTt
= Ho + V,
(7)
Ho = f&h - cc AcogA:A”_,C,‘Cn, ” Q
(8)
v = 1 c Lemnfc,
(9)
”
e=*1
Vnn+e = -Jexp
-c (
4
(A: - A:+e) (b, - bt,& . )
H,, , which arose from applying the transformation
(10)
to the first two terms of eq. (l), now has the simplifying property that the electron and the phonon variables are completely separated, whereas they were completely mixed before the trans-
0. ENTIN-WOHLMAN
562
AND
D. J. BERGMAN
formation. Consequently, the one-electron eigenfunctions of H,, are
where 10) is the electron-phonon vacuum state and 19is an arbitrary function of the phonon creation operators. To obtain the form of these functions in the original representation, we apply the inverse transformation T-l = Tt to (lOa), noting that Tt IO) = IO): T+C,t0 (b;) IO) = T+C,tTT+e (b;) TT+ IO)
The last result justifies the assumption made by Holstein and allows us to identify
dNQ)
a&V,)
=
exp 1 Ai (b, - bt_,) 0(6:) 10). 1 (Q
It also permits us to identify the operator which creates a single dressed electron (i.e., a polaron) at the site n as TtCJT
= exp x Ai (b, - b!_,) CJ. > (4
The second term in H,, represents the binding energy of the polaron. This is the energy needed to localize the electron near the site n and to displace the internal elastic coordinates of the molecules at and around that site. The effect of the unitary transformation Ton H is to cancel the usual electron-phonon interaction [Le., the second term in (l)]. On the other hand. the overlap integrals J acquire a phonon dependence. 2. The occupation probability of a localized state. We now derive an equation of motion for the probability that a localized state is occupied at a time t. The derivation is based upon the formal theories of irreversible processes developed by Zwanzig’) and Sewel13). In these theories averages are defined with respect to the microcanonical ensemble, and consequently the application of such theories to the electron-phonon system is cumbersome. We shall define averages with respect to the canonical ensemble. This modification is not trivial and causes alterations in the basic formalism of the above-mentioned theories. The Hilbert space of the system is spanned by the wavefunctions of H,, [eq. (S)]. We divide this space into N cells according to the eigenvalues of the number
LOCALIZED
STATES VERSUS BAND STATES FOR POLARONS
563
operators CiC,. In this way each cell characterizes a situation in which the electron is localized at a certain site and the phonons are free to be in any of the states e(bb) 10). The part Y [eq. (9)] of the hamiltonian causes transitions among these cells. We define a coarse-grained projection operator 9 which acts upon operators in Hilbert space : 90 = c Tr (SC,tC,,j (l/W,) e-B”pbC,?C,, ”
(11)
W,, = Tr (e -BHphC.tC,).
(12)
When B operates on any operator 0, it replaces it in every cell by the diagonal operator e -BHphCtC n “, suitably normalized so that the trace in that cell is the same as for 0. BO is called the gross part of 8 since it coarse-grains over the details of the microscopic states in each cell. The probability P,(t) that the nth localized state is occupied at a time t is:
p,(t) = Tr (e(t) CJG),
(13)
where e(t) is the density matrix of the system. The evolution in time of P,,(t) is determined by the evolution in time of e(t). To find the latter, we introduce the Liouville operator 8: 9==!?0+8,,
(14)
900 = No, 01,
(14a)
910
(14b)
= [V, U]
and obtain
e(t)= @l(t)+ xe
@z(O)- (i/A) i dt’ (1
-wti)recl-s)t’
0
2
(&
(t
_
where @l(t) = pe (0,
(153)
e&j = (1 - 9) e(t).
(15bj
We assume that at the initial time t = 0: e(O) = C e. Cl/ WJ ewBHphCC,, n
564
0. ENTIN-WOHLMAN
AND D. J. BERGMAN
where the numerical coefficients en give the initial weight of each cell. If en = h,;; then at t = 0 the electron is localized at the llth site, while the phonons are in thermal equilibrium. This seems to be a realistic initial condition, because (as argued in the following) the lifetime of the electron in a localized state is much longer than the characteristic time of the phonon processes. The initial condition implies that: 9~ (0) = ~~(0) = e(O), so ~~(0) = 0. Inserting this into (15) and using (13) we obtain :
dpn (t) = i ’ Tr [C~C,$‘~ (f)] = -$- 1 i dt’ n’ 0
dt [(CJC,IC,)
Tr
x
e-(i,‘h)u(l-s)t’
x (l/W,,> P,,, (t
$p
(,-%h
C$-+)]
- t’).
(16)
To get this result we also used the relations
which can be verified from (11) and (14). Since we have not made any approximations
in deriving
equation for P,(t). In order to solve it, we shall now investigate the kernel. We write this kernel in the form: (I//PW,.) I(O)(f)
=
nn’
~$(t’)
Tr [9 (CiC,J e-(i’A)r(l-@)t’
-l-r
[(y
C 1
= (-i/h)’
n+
C
n
)
e-“‘“‘zO”
g2cr)
=
e-‘i”‘“O”
2,
of
(17)
(17a)
(tl) ..* _Y2(fI)
(1
_
uj3)
I#
0,
e-ci~fi)oOre
1 ((Vn’n’+E(f’) V”,,,,.) &=+l
(17b) (17c)
We now calculate the first few orders of the kernel The zeroth order, from (8)-(IO), (14) and (17a) is:
r;;?(t’) = (l/P)
= t I$ (t’), 1=0
(ewBHphCJL$,)] (I/ W,A”),
9,
(ePBeph CJCn,)] (l/W,A’),
eWfi).LPOf 6?p,
it is an exact
the properties
11-l ...j dt,
j’dt,
x Tr [(~,C~C,,) x 2,
B (eePHph C$C,,)]
(16)
and discuss the general
+ cc.) (&I”*+, - &“,).
term.
(18)
LOCALIZED STATES VERSUS BAND STATES FOR POLARONS
565
Here (O,,), where ODhis a function of phonon operators, is the equilibrium and O,,(t) = e”‘fi’HDh’ OPhe-(l’fi)H@‘r.In order average : Tr (e -BHnhO,,J/Tr (eWBBph) to calculate higher orders we note that:
~&> 0 = VW, 01 - c O/K,) Tr (OC~IG,)MO, ewBHph C.‘,G,l. “1
(19)
The first-order term is: 1’l! nn (t’ 3 tl) = -(i/A” W II’) Tr (e -BHDhCX”,
[K WI),
[W’),
CGlll>.
Explicit calculation of the commutators yields that this expression consists of 8 correlation functions of VnnLt 1 of the type : E BC+l(V~,“,+rVn’+~“‘+d+E’ 01) v”,+e+el+“4’))4+,,,*1~““.. 9,
-
Since E + &I can be either zero or +2 these correlation functions vanish. It is easy to see that, in general, any odd-order term of the kernel includes a delta function which makes the whole expression vanish unless the sum of an even number of E , each of which is 5- 1, is equal to + 1. Since that is impossible all the odd-order terms of the kernel vanish. The second-order term is :
1;;) (t’tlt*)
= -(l/PWn,)
x Tr (ewBHDh C$G W, [J%), [WA VW>, C,!C,llll> +-
l PW,.
CL
Tr (emcHphC,$C,,, [V, [ V(t,), C,?lCn,]])
Ill w,,
x Tr (eeBHph Cn’,G,PW, WV), C!C.ll> and consists of 16 correlations of V4 minus 16 products of two V2 correlations. A typical V4 correlation is:
0. ENTIN-WOHLMAN
566
AND
D. J. BERGMAN
where we have defined: ~“‘“2 E A;’ - A;’ 4
(20a)
and K.
E
(( Vn,+,))2 = J2 exp
A2 -- NMfi
1 - cos q” coth t?!!. c4 UJ3 2 g
.
(21)
>
Kt decreases exponentially when T increases and is always smaller than J’). We need to know the behaviour of the correlations at long times. This behaviour is found by the method of steepest descent which gives:
x cos
[(cog- co:)”(tl - ti + C0, (fl - I,)+
$$A) 4 +7r]
(22)
Thus, if any of the time differences in (20) is much larger than I/C!), , the suitable sum in the exponent tends to zero. This decay property entails that whenever the difference between the times (t’, ti) and (f2, 0) is larger than l/to, the V4 correlation is decoupled into a product of two V2 correlations. This does not imply, however, that I,$) (t’, tl, t2) tends to zero. We have listed the diagrams which remain in Zii? (t’, tl , t2) in fig. 1. The correlations that are described by these diagrams tend to Kg when all the time differences in (20) are larger than l/~,j,. It is clear that these diagrams would appear even if one were to calculate the second-order term in a system without any electron-phonon interaction. In such a system A = 0, T = 1 and Vnn+C = -.Z, so that the second-order terms would be given by the diagrams of fig. 1, where each diagram is now multiplied by J4 instead of Kz. It will be shown in the next section that 4K$ is indeed the bandwidth of the polarons. The diagrams which remain when all the time differences are much larger than l/w, describe the virtual-phonon processes which manifest themselves in the temperature dependence of the bandwidth. The calculation of Z,‘f!(t’, t, , . . . , tL) for an even I > 2, is quite straightforward, although cumbersome. The Ith order consists of 2’+’ correlations of order V+ 2, minus all their decouplings into products of V2, V4, . . . , V’ correlations, all of which involve sequential times. But again, even when the longest time t’ is much greater than I/o,, the Ith order does not vanish. This failure of the kernel to vanish even for times much greater than l/o, which appears in every order, is worthy of note. In the general theory developed by Sewel13) it was assumed that
LOCALIZED
STATES VERSUS
BAND STATES FOR POLARONS
567
if after a rather short time (Le., short compared to the lifetime of a state) the correlations decouple into products of lower-order correlations, the kernel vanishes. In the next section we will see that this vanishing property does hold for the bandstates representation.
VW
”
n’-4anI+, Ws)
v(t,)
VU.)
Y
n8-f~nB+4
n”-t~n~+9 VW )
nbmn1+2
VW)
&2-n* 4
VW)
602)
v(t,
1 If1 v(t’)
v(t, 1
n’c2Qno+2 ”
2
Fig. 1. The diagrams which describe the correlations that remain in I,$f? (t’, t,, tz) when I’, tt and t2, 0 are separated by an interval % l/o1 . The vertices indicate lattice sites and an arrow accompanied by V(t) between two vertices n, n + E represents the operator V,,,+,(t) C.‘C.+.. The correlation described by the first diagram is: < V,,,,,,_i I’,,._,,, (ti) I&,+i(t’) &f+ln,(tz)), and it appears three times in I,$f? (t’, tl , tJ because in this diagram n can be equal to n’, n’ k 1.
From eqs. (20) and (22) it can be seen that the second-order term is negligible compared to the zeroth order if the following conditions hold: J/fic0, < 1,
(23)
K$‘/k < 1.
(24)
The reason for this is as follows: any sum over q in the exponent of eq. (20) differs from zero as long as the time difference appearing in it is B 1I@,. We approximate the sum in this case by its maximum value, attained when the time difference in zero. Thus, that part of the exponent together with K$ gives J. When the time difference is > l/o,, the sum tends to zero [eq. (22)]. Therefore, in this region the exponential is approximated by 1. The same arguments hold for the I > 2 orders of the kernel. Consequently, when (23) and (24) hold, all the 1 > 0 orders
568
0. ENTIN-WOHLMAN
AND D. J. BERGMAN
of the kernel are negligible compared to the zero order. The physical meaning of these conditions will be discussed later. Under these conditions the equation for P,(t) [from eqs. (18), (20) and (21)] is:
dpn Ko = $y egkI(Bnn.+, - b,,.) dt
f dt’P,,s (t -
A2 (1 -cos
qu) cos ~0~(t’ + _ti@)
NM&o,3 We now write the term in the brackets
(
exp C [
A2 ;;;;;,a)
t’)
0
sinh &%%x,
..1.
+ cc >
on the r.h.s. in the form:
cos;;;;;
+ C.C. - 21 + 2
Q
and divide it into two parts. The second part would have remained even if there were no electron-phonon interaction (multiplied by J2 instead of Ko). It thus bears the influence of the virtual-phonon processes, and consequently contributes to changes of P,(t) due to tunnelling. The first part vanishes for w,t’ 8 1 [see eq. (22)] and describes phonon-activated processes. If we consider P,(t) at times that satisfy:
the contribution
of the second
term to dP,,ldt becomes
markoffian,
and we ob-
tain
dp,,
-
dt
=,=j,
-
(4m,+,-
S,,,)
K,P,,
2Ko
(t) + -
82
(26)
where (27) -CO
It is easy to see [by changing the variables in eq. (26): K,t = t; P,(t) = p,,(7)] that the relative importance of the two terms on the r.h.s. of eq. (26) is determined by the factor K$X, . It is shown in the following that Kl is the transition probability per unit time for a hopping process. Thus, Ki/fiK, is the ratio of the transition probability of a tunneling process to that of a hopping process. K. is an exponentially decreasing function of T which attains its maximum value at T = 0 [see eq. (21)], while Kl vanishes at T = 0 and increases as the temperature is raised:
KoU’J13/W (Td > CKo(G)13/%(TA
T, < T2.
LOCALIZED STATES VERSUS BAND STATES FOR POLARONS
569
Therefore, at sufficiently high temperatures which satisfy
the hopping mechanism dominates over the tunneling mechanism. That is, the second term on the r.h.s. of eq. (26) is negligible compared to the first and consequently, P,(t) N eezKlt 1,-1 (2K,t),
(29)
where I,,(X) is a modified Bessel function. Here we assumed that P. (t = 0) = a,, 1. [To solve eq. (26) when the second term is negligible we Fourier-transformed it, solved for the Fourier transform, and then found the inverse Fourier transform of that solution.] From (29) we see that in the temperature region in which (28) holds, K;l is the characteristic lifetime of a localized state, and Kl is the transition probability per unit time for a hopping process. The transition probability increases with T because the number of accessible phonons increases with T and thus the phonon-activated processes become more frequent. At temperatures higher than the Debye temperature, Kl assumes the form of a thermal-activated transition probability’*4). We now examine the conditions under which the solution (29) was obtained. The first condition, eq. (23) is temperature independent and determined by the physical properties of the system. It means that the tunneling frequency J/hof the bare electron must be smaller than the characteristic frequency of phonon processes. The second condition, eq. (24), reduces to eq. (28) when t’ is replaced by the lifetime K;l. Thus, a consistent solution is obtained in the range of temperatures determined by (28), in which the hopping mechanism dominates over the tunnelling mechanism. When eq. (28) holds, the second term on the r.h.s. of (26) is negligible compared to the first term, and eq. (26) becomes markoffian. Since in order to obtain (26) we used eq. (25) co,/K,% 1 must hold for a consistent solution. Namely, the lifetime of a state must be much longer than the microscopic time l/w, which is the lifetime of the integrand in Kl [eq. (27)]. Thus, both conditions w,/K,% 1 and K$/AK,4 1 are needed for the markofficity of the equation for P,(t), and they determine the temperature region in which the representation by localized states is adequate. In the temperature region in which the tunneling mechanism dominates over the hopping mechanism [i.e., when eq. (28) is violated] the first term on the r.h.s. of eq. (26) is negligible compared to the second, and the characteristic time of P,,(t) is of the order A/K$. Going back to (24) we see that the I > 0 orders of the kernel are not negligible compared to the zero order. If we neglect the realphonon processes, that cause the hopping, in the kernel and keep only the virtual. phonon processes, the order of magnitude of the Ith-order terms of the kerne [eq. (17)] is (K,,/A') (Kt f/h)'. Thus, the characteristic time of the kernel is the
570
0. ENTIN-WOHLMAN
AND D. J. BERGMAN
same as the characteristic time of P,(t), i.e., both are equal to the tunneling time A/K;. Consequently, the equation for P,,(t) is not markoffian. Note that solving for P,(t) in this case is equivalent to solving the equation in the absence of the electron-phonon interaction. We show in the next section that in the region of low temperatures where Klf/#iK, > 1 the appropriate representation is by band states. 3. The occupation probability of a band state. In order to derive an equation for the evolution in time of the occupation probability Pk(t) of a band state k, we must construct the phase cells in such a way that each cell represents a state in which the electron is in a state k, while the phonons are in any of the states 8(bi) IO). We first Fourier-transform the hamiltonian (7) to obtain: H=H,+H,,
(30)
H, = HPh + c (ek - Eb) C,tC,,
(31)
k
&k
=
2K$ cos ka,
(314
(31b)
E,, = 1 hq,A;A)tq,
H alk2 = (l/N)
c n
c
(Vnnfe - K,f) e-iol-kZ)nufi’@,
(324
&=kl
where we have defined: C,t = (l/N”) T eeikna C,t
and k lies in the first Brillouin zone of the one-dimensional lattice. The vector space of the system is now spanned by the eigenfunctions of H,, and it is divided into coarse-grained cells according to the eigenvalues of the number operators c,tCk. Thus, C,tck is the projection operator on the kth cell. The part H, of the hamiltonian does not cause transitions among the various cells. It follows from (31a) that 4K$ is the bandwidth of the electrons when the electron-phonon interaction is taken into account, i.e., it is the bandwidth of the dressed electrons* l*“). The probability that a band state k is occupied is Pk(t)
=
Tr
[e(t)
c,tckl.
* In ref. 5 the polaron bandwidth in a three-dimensional lattice was calculated iational method. Kt is the one-dimensional version of that result.
(33) by the var-
LOCALIZED STATES VERSUS BAND STATES FOR POLARONS
571
Running through the same calculations as in the last section we obtain the same equations as (16) and (17) except that H,,, V, Cf;, C, are now replaced by K, Hi, CL Ck. The calculation of the orders of the kernel in the equation for dPk (t)/dt is carried out as in the previous section. In the present case, correlations of an odd number of Hklkl do not vanish. The Ith-order consists of 21+2 correlations of (I + 2) Hklkz, minus all their decouplings into products of Htlk2, H&, .*. Hk,,+ correlations, where all the correlations involve sequential times and each HklkZ(t) introduces a factor e(i’fi)(ek~-ek~)fdue to the time dependence of Cl,C,,. The important difference between the kernel in the equation for dP,/dt and the kernel in the equation for dP,/dt is found when t’ > l/o,. For these times all the orders of the present kernel tend to zero. We exemplify this behaviour by considering the second order of the kernel. It consists of 16 terms of the form:
(fbklHklk2 (t2)H&d g
exp
{(i/h)
[t&k1
02))
-
&k’)
H/&t’)) t2
+
c&k’
H,,,(f)))
-
&k)
ttl
-
t’)l)
*
When t2, tl < l/w, the time t’ is separated from all the other times including the time 0 and consequently [see eq. (22)] H&&t’) becomes uncorrelated with the other P&k,. Since (H,&> = 0 [from eq. (32a)] the two terms in eq. (34) vanish. Similarly, when t’, t2, t, > l/m,, the time 0 is separated from all the other times and again (34) vanishes. When t’, tl ‘> l/o, and 0, t2 < l/w,, the first correlation is decoupled and becomes: (Hk’kl HklkZ(tz)>
(35)
where t is the lifetime of a band state. If this condition is violated, then the uncertainty in the energy of a single band state will exceed the bandwidth and the band picture is certainly inadequate. From (35) and the fact that J < ho, [see
572
0. ENTIN-WOHLMAN
eq. (23)] and K$ < J it already olz
> Jr/A > Kt+i
i.e., the condition
AND D. J. BERGMAN
follows
that
> 1,
(36)
for markofficity
is satisfied.
As a result of this, the exact equa-
tion for dP,ldt, y
= $dt’;
Pk> (t - t') GkkQ’),
[where Gkkf(t’) represents the kernel (17) when H,, HI, Ci, Ck, respectively] can be approximated tion = ; P,,(t) j%G,,,
y
H,, V, CA, C, are replaced by by a markoffian master equa-
(f’).
The upper limit of the integral was taken to co because of the decay property of the kernel. The Zth order of the kernel is of the order (K,/Fz’) (J/ho,)‘, thus from eq. (23) the I # 0 orders are negligible compared to the zeroth order. Note, however, that it is not necessary to have J < ko, in order to obtain markofficity in the band representation. If J 3 fro, the calculation of the kernel merely becomes more complicated as all orders then have to be taken into account. Keeping only the zeroth order, the master equation is
dP, (0 = _
eBp(W -&*) Pk,(t) -
5 [akk,
mkjk
eiB (i’n-‘x’) Pk(t)].
(37)
dt where
cc
(i/A) (t&s
x jdte -
x
Cl
-.Qt
CL
_
emla)
(1
_
e-igea
‘OS @I+ sinh $3ri~, >
The solution
of eq. (37) can be written
k’#k
1
1 .
(374
in the form:
Pk(t) = emxkf P, (0) + J’dt’ 1 AkkpPkT (t - t’) eexkr’, 0
_
(38)
LOCALIZED
STATES VERSUS BAND STATES FOR POLARONS
573
where xii = c Aktk,
(384
k’itk
A kTk = e - fS(W -Ed
&k’k*
Wb)
In order to find the lifetime of a band state k we rewrite the differential equation for Pk(t) in the following form dP, dt
= -XkPk + c &‘Pk’.
(38c)
k’fk
From (37a) it can be shown that all the A kk’ that we need are of the same order of magnitude (in fact, all of them are w l/N), and thus that
Since c Pk = 1, the second term on the r.h.s. of (3%~)is like an average value of Akk,, and is therefore also of the order 0 (l/N) compared to xk. Consequently, so long as Pk = O(l), we can neglect the second term and write dP,/dt % -XkPk. Thus the decay time of Pk is l/xk. In order to evaluate xk we first write akkJ in the form Olkk,=
(~O/h2)
;
eWfi)(e,~
-%)t
(Z/N)
1
--oo
eUk-k’ha
n
x [(eLn(‘) _
1) +
(esLnct) - l)cos(k
Na L,(t)
=
4AZ2 _!!$ ~0s
Mfi
nqa
sin2
x s ma
25
+ k’)a],
‘OScogt
2 sinh $3tZw,’
(39) (394
0
By inserting (39) into (38a) we obtain: xk
=
(2~,/h2)
j
dt
e-(*/h)ek(t+fiSh)
-cc
x C
”
eikna(i)” (eL’.“’ - 1) J, (z$(,+!E))
+ + (e-W)
I _ l)[e+V.,,
- eiko J ”_ l(2+yt+y))]}.
(?(I
+ T))
(40)
574
0. ENTIN-WOHLMAN
AND D. J. BERGMAN
We change t + $i/?fi -+ t and make use of the property L,, (t + $/?A) -+ 0 as o,t B 1 (this can be seen by the method of steepest descent): ,yk
=
(2&/fi2)
7 dt
e-ci’kr/fi)1 eikna(i)”
Because for o,t > 1 the integrand tends to zero, the important contribution to the integral comes from the times w, t d I. From (23) we see that in this region the arguments of the Bessel functions are very small : (Ko)/fi) t < (J/h) t < co,t < 1 and e -i(Eht/fi) tends to 1. Therefore xk c (2K,/ti2) By averaging cxk)
=
i dt [eLoCt)--co
this expression
we can write xk as follows: 1 + sin ka (edL1”’ -
I)].
over k we find (41)
2K,,
where K, was defined in eq. (27). Therefore, the mean lifetime in a band state is of the order KL’. The mean lifetime in a band state increases as the temperature decreases, and at T = 0 it tends to infinity. I.e., at zero temperature the band states become the eigenstates, as expected. Inserting z z KY’ into eq. (35) we see that the band representation is appropriate at low temperatures. From eqs. (28) (35) and (41) it now seems that at the temperatures
which satisfy
both the band picture and the localized-states the adequate representation would probably over a finite number of lattice sites.
picture are unsuitable. In this region be in terms of states which are spread
4. Discussion. We have presented a method for finding the suitable representation of states as a function of the temperature for a model system with strong electron-phonon interaction, As this model forms one of the few examples in which the formal theories of irreversible processes can be worked out, it enabled
LOCALIZED STATES VERSUS BAND STATES FOR POLARONS
575
us to examine some of the assumptions and approximations which are made in these theories. In section 2 we presented a modified derivation of the equation for the occupation probabilities, in which averages are calculated using the canonical ensemble. In the case of localized states we have found that although the two products V(t,) .=a V(t,) V(t’) and V(t,+,) ... V(t,) V become mutually uncorrelated whenever their respective temporal ranges (t,, t’) and (0, t,, 1) become separated by more than l/w,, the equation for the probability does not become markoffian. This possibility was overlooked in ref. 3. Eq. (16) for the probability J’,(t) and the analogous equation for Pk(t) were derived exactly, without any coarse-grained approximations. It is only when one tries to calculate averages using P,(t) that one makes coarse-graining approximations. That is, one approximates the average of any operator 0
01,
by using the coarse-grained part PI(t) = SQ (t) in place of e(t):
where 0, is the equilibrium average of 0 in the nth cell: 0, = (l/ W,,) Tr (emDHph OC,?C,). Since it is impossible to solve the exact equations for the occupation probabilities, we found markofficity conditions for each equation. We then solved the approximate markoffian equation to obtain the temperature-dependent lifetime of a state and to find the temperature region in which the representation is adequate. In the region near the transition temperature [given by eq. (42)], neither representation is adequate. Therefore in this region neither the band states nor the localized states form a good approximation to the true eigenstates of the system. The problem of the proper intermediate representation can also be solved by our method. Assuming a certain representation for the states in the intermediate region, the phase space should be divided into cells according to these states. Then the exact equation for the temporal evolution of the occupation probability can be found. The decay properties of the kernel in this equation will yield the markofficity conditions for the equation. If the markoffian equation can be solved to give a lifetime which is consistent with these conditions, and if it is possible to prepare the system in the proposed states, then the representation is an adequate one. Acknowledgements. many helpful discussions.
One of us (O.E.W.) would like to thank P. Gluck for
576
0. ENTIN-WOHLMAN
AND D. J. BERGMAN
REFERENCES 1) 2) 3) 4)
Holstein, T., Ann. of Phys. 8 (1959) 343. Zwanzig, R.W., Boulder Lectures in Theoretical Physics (1960). Sewell, G.L., Physica 34 (1967) 493. Entin-Wohlman, O., Ph.D. thesis, unpublished (1972). Clark, P.M.. Gluck, P. and Wohlman-Entin, O., J. of stat. Phys. 7 (1973) 361 5) Sewell, G.L., Phil. Mag. 36 (1958) 1361.