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Incorporation of finite temperature structure and statistics into conductivity calculations in the Coulomb gap
Volume 151, number 3,4
PHYSICSLETTERSA
3 December 1990
Incorporation of finite temperature structure and statistics into conductivity calculations in the Coulomb gap A. H u n t Physics Department, University of California, lrvine, CA 92717, USA Received 22 May 1990;revised manuscript received 10 August 1990;acceptedfor publication 28 September 1990 Communicatedby A.A. Maradudin
We present a new theory for de transport in the Coulombgap. The theory differs primarily from that of Efros in its physical interpretation. It is useful, however, because it yields as well analytical expressions for the pre-exponential dependenceand because it demonstrates that many-electronhoppingis irrelevant, settling a long-standingdebate.
A recently developed theory [ 1 ] gives a thermodynamic treatment of the electrons in the Coulomb gap [ 2 ]. In this theory it is shown that in systems of arbitrary dimensionality domains of radius rooc ( k T ) - ~ exist, in which the electrons are ordered antiferromagnetically. The result is obtained by minimizing the free energy of the electrons within the Coulomb gap. It is shown here that this result is of crucial importance to transport. First we review briefly the calculation of the free energy, and then we discuss the situation in transport. In the model, only those electrons within the Coulomb gap are treated. The width of the Coulomb gap is determined by a competition between disorder energy and Coulomb interaction energy [ 3 ]. For charge neutrality, a positive charge of ½e is associated with every site, an electron with half of the available sites. In each domain, the electrons are assumed to exist in one of two possible states, obtainable from one another by exchanging electrons for holes. The entropy of each domain is therefore k In 2. The entropy of the system is then
S = ( V/ Vo)k ln 2 = ( V/rao)k ln 2 ,
(1)
where V is the (generalized) volume of the entire system, and Vo=r d is the volume of each domain. The internal energy of the system is given by a Ma-
delung sum, less a remainder term, which has been shown to be proportional to r6"Ca+t), i.e.
U= - E g [ 1 - 2 ( r J r o ) d+' ] V/r d ,
(2)
where the width, E~, of the Coulomb gap is given by Eg= e2Nm/erg, Nm is an appropriate Madelung sum, rg is the average separation of, and V/r d the number of, electrons in the Coulomb gap. In the above expression, e is the electronic charge, and ~, the dielectric constant. Forming
:~= U - T S =-Eg[1-2(q/ro)d+~]V/rg-kTln2
V/rao (3)
and optimizing with respect to ro yields
ro o c r g E J k T
(4)
and
~oc - E g [ 1 + ( k T / E g ) d+ ' ] .
(5)
The states at domain endpoints (l-d), edges (2-d), or corners (3-d) may easily be seen to have zero interaction energy, as the other charges are arranged exactly antisymmetrically about these domain extremities. Thus the total number of single electron states within an energy k T of the Fermi level is immediately seen to be proportional to rffdoC k T d. Con-
0375-9601/ 90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
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Volume 151, number 3,4
PHYSICSLETTERSA
sequently, the finite temperature density of states at the Fermi energy is rffd/kT, or
N(T) pcrgd( k T/ Eg)d- 1 .
(6)
We also obtain the density of states at energy E (for T= 0) by differentiating with respect to T and evaluating at E, N(E)oc
(~/e2)dE d-I .
(7)
We note that the latter result (eq. (7)) is in agreement with that of Efros [ 2 ], and the former result (eq. (6)) with ref. [2] (where it was obtained by a generalization of the Efros method as well as by other methods) and with Efros [ 4 ]. Another interesting result relates to the degeneracy of the many-electron states with electrons at domain boundaries (with zero interaction energy). The result will be most apparent in one dimension. The hop of an electron from a domain boundary to a site one half the typical electron separation to the left merely transposes the domain boundary one full electron separation to the left, and the two manyelectron states are degenerate, to a good approximation. Thus electrons at the boundaries of domains are in general much more mobile than electrons in the middle of domains, which need an activation energy on the order of the Coulomb gap in order to be transported an appreciable distance. As a consequence, most transport occurs on domain boundaries. On the domain boundaries, however, the density of states is shown below to be a constant. For transport along the domain surfaces in a d-dimensional system, we can therefore use the Mott [ 5 ] variable range bopping argument for the dc conductivity, but with a temperature dependent density of states at the Fermi level. The energies of the states in the, say, two-dimensional surfaces bordering three-dimensional domains are to a good approximation independent of the ordering of the domains. This is because the domains must be oppositely ordered on opposite sides of the boundary in order to define the domain boundary. This antisymmetric electron orientation with respect to the domain boundary means that electrons within the domains contribute zero interaction energy to the energy of the electrons on the domain boundaries. As a consequence, the density of single electron energies (DOS) for states con188
3 December 1990
strained to lie in these surfaces is identical to the result for the two-dimensional DOS. Further boundaries lying within these two-dimensional surfaces produce one-dimensional regions of space, whose energies are also independent of the remainder of the two-dimensional surface. Thus a fraction 1~to of the 3-d sites lie in the surface with N(Ef) oc T, consistent with N(Er) oc T/ro, oCT 2 for 3-d. Moreover a fraction l/r 2 lie on one-dimensional paths with N(Ef) temperature independent, so that N(Ef) oc 1?2 oc T 2 once again. The problem is now the following. If electron excitations are restricted to originate and wind up on the 2-d domain boundaries, but no restriction is made on which boundaries, the number of excitations within radius r is proportional to r 3, as the boundaries are embedded within a three-dimensional space. The DOS which applies, however, is proportional to T 2 anyway, because the restriction to the surfaces with two-dimensional DOS (proportional to T), introduces the second factor T. Thus a restriction to the two-dimensional surfaces is indistinguishable from a straightforward bulk application of the Mott optimization procedure. But applied to the three-dimensional DOS, this procedure yields 1
d~dr__drr\( 2r_~ r3N (_Ef) k T ) = 0
(8)
and ~optpC( Tol T) 3/4 ,
(9)
which is wrong! The same applies if one treats only the one-dimensional paths, as long as the hopping is not constrained to connect initial and final states on the same 1-d path. But if the hops are so constrained, we can use the l-d density of states, i.e. N(Er)pc T °. At the same time, the number of states encountered within a distance r is proportional only to r (and not r 3) so that, ¢=2r_~
a
1
rTO(kT ) .
(10)
Optimization yields
Ropt =Ro exp(~p,) =Ro e x p ( T o / T ) ' / 2 ,
( 11 )
which leads to a larger conductivity than in eq. (9). Moreover, this result is independent of the dimensionality of the system, and is in agreement with that
Volume 151, number 3,4
PHYSICS LETTERS A
of Efros not only with regard to the exponent, but also with regard to the value of To which is given by eE/~ak. Thus one-dimensional paths define completely the current carrying network in any dimension, and a single electron percolation theory can be justified (further justification is given below). We can also calculate the pre-exponential. It is proportional to the separation of the transitions with the slowest rates, which scales with to, and inversely proportional to the areal density of paths. The areal density of paths is r~ -2 in three dimensions. Thus,
e 2 Pph k T ad¢- k T r, -~g e x p [ - ( T ° / T ) ' / 2 ] =~Uph exp[ -- (To~T) 1/2 ]
(12)
in three dimensions, while e2
ad¢ = ~-~ Uphexp[ - (To~T) 1/2 ]
3 December 1990
One final point regarding transport should be mentioned. Sequential correlations are not expected to modify the above result for the dc conductivity, as all quantities are treated in thermodynamic equilibrium. The low frequency ac conductivity, however, may be strongly affected by sequential correlations, as it is clear that the relaxation times of any non-local relaxation processes will depend strongly on whether the hops are sequentially correlated. A future publication will deal with this topic. An interesting question is: How long does it take, say, for eight domains of size r0 to coalesce to form one domain of size 2ro, when the temperature is dropped by a factor two? If many-electron transitions were required (for reasons of energy) such a transition could require a time proportional to toc exp [ (2ro/rg) 3rg/4a] oc exp [ (Eg/kT)
(14)
(13)
in two dimensions. An important result is that transport takes place along strictly one-dimensional paths, whose density of states if given by one-dimensional results. In l-d the importance of Coulomb interactions can be easily neutralized by disorder, as the total Coulomb energy and the total disorder energy both depend linearly on the concentration of the electrons. In two and three dimensions the Coulomb interaction energy is a, sublinear function of the concentration guaranteeing that the electrons near the Fermi level feel the Coulomb interaction even in the limit of large disorder. This result could explain why the existence of the Coulomb gap has not been conclusively demonstrated through transport measurements. As a further observation we mention that, as no Coulomb barriers are encountered (on account of the degeneracies along the domain boundaries discussed, and on account of the finite density of states at E = E f ) multi-electron transitions need play no role in overcoming such barriers. Moreover, it has been shown [6 ] that correlated many-electron probably plays no role in transport already in two dimensions, so that in one dimension no effect should be seen. As a consequence (in contrast to previous claims by the author [ 7 ] as well as by Pollak and Knotek [ 8 ] ) no particular effect on the dc conductivity due to multielectron transitions is expected.
3rg/4a],
which can be enormous for k T m u c h smaller than Eg. Single-electron transitions could require (at most) a time toc exp (Eg/kT)
( 15 )
each, while the sequential correlation of such transitions could at most extend the time to
toc (Eg/kT) 3 e x p ( E J k T ) ,
(16)
which is obviously much smaller than eq. (14) for Eg >> kT. Thus many-electron transitions appear irrelevant for the approach to thermal equilibrium as well, as result which would explain the relative success of the computer simulations of Davies et al. [ 9 ] and Gruenewald et al. [ 10 ]. The fact, however, that r0 diverges in this limit T-,0, suggests that the DOS obtained by these authors in the limit of zero energy and at zero temperature is unreliable. And in fact, in the limit of zero energy, computer simulations have been unable to distinguish between the quadratic gap and other harder gaps which have been proposed. Moreover the fundamental difference between the system at T = 0 and at any finite temperatures makes questionable any approach to transport based on zero temperature statistics. As a final comment on thermodynamics we point out that the Efros density of states has often been treated as an established fact, but it is strictly speaking an upper limit. It is tempting to argue that the 189
Volume 151, number 3,4
PHYSICS LETTERS A
density o f states given here is a lower limit, as in a real system the " f e r r o m a g n e t i c " o r d e r i n g is n o t expected to be perfect. Since the lower l i m i t a n d the u p p e r l i m i t are in exact agreement, it is felt here that this q u e s t i o n has also b e e n settled once a n d for all.
References
[ 1] A. Hunt, submitted to Philos. Mag. Lett. (1990). [ 2 ] A.L. Efros, J. Phys. C 9 (1976) 2021.
190
3 December 1990
[3] A. Hunt and M. Pollak, J. Phys. C 18 (1985) 5325. [4 ] A.L. Efros, private communication (1990). [5] N.F. Mort, Philos. Mag. 19 (1969) 835. 16] R. Chicon, M. Ortuno, B. Hadley and M. Pollak, Philos. Mag. B 58 (1988) 69. [7] A. Hunt, J. Phys. C 20 (1987) 1469. [8] M. PoUak and M.L. Knotek, J. Non-Cryst. Solids 32 (1979) 141. [9]J. Davies, P.A. Lee and T.M. Rice, Phys. Rev. Lett. 49 (1982) 758. [ 10] M. Gruenewald, B. Pohlmann, L. Schweitzer and D. Wuertz, J. Phys. C 15 (1982) 1153.
PHYSICSLETTERSA
3 December 1990
Incorporation of finite temperature structure and statistics into conductivity calculations in the Coulomb gap A. H u n t Physics Department, University of California, lrvine, CA 92717, USA Received 22 May 1990;revised manuscript received 10 August 1990;acceptedfor publication 28 September 1990 Communicatedby A.A. Maradudin
We present a new theory for de transport in the Coulombgap. The theory differs primarily from that of Efros in its physical interpretation. It is useful, however, because it yields as well analytical expressions for the pre-exponential dependenceand because it demonstrates that many-electronhoppingis irrelevant, settling a long-standingdebate.
A recently developed theory [ 1 ] gives a thermodynamic treatment of the electrons in the Coulomb gap [ 2 ]. In this theory it is shown that in systems of arbitrary dimensionality domains of radius rooc ( k T ) - ~ exist, in which the electrons are ordered antiferromagnetically. The result is obtained by minimizing the free energy of the electrons within the Coulomb gap. It is shown here that this result is of crucial importance to transport. First we review briefly the calculation of the free energy, and then we discuss the situation in transport. In the model, only those electrons within the Coulomb gap are treated. The width of the Coulomb gap is determined by a competition between disorder energy and Coulomb interaction energy [ 3 ]. For charge neutrality, a positive charge of ½e is associated with every site, an electron with half of the available sites. In each domain, the electrons are assumed to exist in one of two possible states, obtainable from one another by exchanging electrons for holes. The entropy of each domain is therefore k In 2. The entropy of the system is then
S = ( V/ Vo)k ln 2 = ( V/rao)k ln 2 ,
(1)
where V is the (generalized) volume of the entire system, and Vo=r d is the volume of each domain. The internal energy of the system is given by a Ma-
delung sum, less a remainder term, which has been shown to be proportional to r6"Ca+t), i.e.
U= - E g [ 1 - 2 ( r J r o ) d+' ] V/r d ,
(2)
where the width, E~, of the Coulomb gap is given by Eg= e2Nm/erg, Nm is an appropriate Madelung sum, rg is the average separation of, and V/r d the number of, electrons in the Coulomb gap. In the above expression, e is the electronic charge, and ~, the dielectric constant. Forming
:~= U - T S =-Eg[1-2(q/ro)d+~]V/rg-kTln2
V/rao (3)
and optimizing with respect to ro yields
ro o c r g E J k T
(4)
and
~oc - E g [ 1 + ( k T / E g ) d+ ' ] .
(5)
The states at domain endpoints (l-d), edges (2-d), or corners (3-d) may easily be seen to have zero interaction energy, as the other charges are arranged exactly antisymmetrically about these domain extremities. Thus the total number of single electron states within an energy k T of the Fermi level is immediately seen to be proportional to rffdoC k T d. Con-
0375-9601/ 90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
18 7
Volume 151, number 3,4
PHYSICSLETTERSA
sequently, the finite temperature density of states at the Fermi energy is rffd/kT, or
N(T) pcrgd( k T/ Eg)d- 1 .
(6)
We also obtain the density of states at energy E (for T= 0) by differentiating with respect to T and evaluating at E, N(E)oc
(~/e2)dE d-I .
(7)
We note that the latter result (eq. (7)) is in agreement with that of Efros [ 2 ], and the former result (eq. (6)) with ref. [2] (where it was obtained by a generalization of the Efros method as well as by other methods) and with Efros [ 4 ]. Another interesting result relates to the degeneracy of the many-electron states with electrons at domain boundaries (with zero interaction energy). The result will be most apparent in one dimension. The hop of an electron from a domain boundary to a site one half the typical electron separation to the left merely transposes the domain boundary one full electron separation to the left, and the two manyelectron states are degenerate, to a good approximation. Thus electrons at the boundaries of domains are in general much more mobile than electrons in the middle of domains, which need an activation energy on the order of the Coulomb gap in order to be transported an appreciable distance. As a consequence, most transport occurs on domain boundaries. On the domain boundaries, however, the density of states is shown below to be a constant. For transport along the domain surfaces in a d-dimensional system, we can therefore use the Mott [ 5 ] variable range bopping argument for the dc conductivity, but with a temperature dependent density of states at the Fermi level. The energies of the states in the, say, two-dimensional surfaces bordering three-dimensional domains are to a good approximation independent of the ordering of the domains. This is because the domains must be oppositely ordered on opposite sides of the boundary in order to define the domain boundary. This antisymmetric electron orientation with respect to the domain boundary means that electrons within the domains contribute zero interaction energy to the energy of the electrons on the domain boundaries. As a consequence, the density of single electron energies (DOS) for states con188
3 December 1990
strained to lie in these surfaces is identical to the result for the two-dimensional DOS. Further boundaries lying within these two-dimensional surfaces produce one-dimensional regions of space, whose energies are also independent of the remainder of the two-dimensional surface. Thus a fraction 1~to of the 3-d sites lie in the surface with N(Ef) oc T, consistent with N(Er) oc T/ro, oCT 2 for 3-d. Moreover a fraction l/r 2 lie on one-dimensional paths with N(Ef) temperature independent, so that N(Ef) oc 1?2 oc T 2 once again. The problem is now the following. If electron excitations are restricted to originate and wind up on the 2-d domain boundaries, but no restriction is made on which boundaries, the number of excitations within radius r is proportional to r 3, as the boundaries are embedded within a three-dimensional space. The DOS which applies, however, is proportional to T 2 anyway, because the restriction to the surfaces with two-dimensional DOS (proportional to T), introduces the second factor T. Thus a restriction to the two-dimensional surfaces is indistinguishable from a straightforward bulk application of the Mott optimization procedure. But applied to the three-dimensional DOS, this procedure yields 1
d~dr__drr\( 2r_~ r3N (_Ef) k T ) = 0
(8)
and ~optpC( Tol T) 3/4 ,
(9)
which is wrong! The same applies if one treats only the one-dimensional paths, as long as the hopping is not constrained to connect initial and final states on the same 1-d path. But if the hops are so constrained, we can use the l-d density of states, i.e. N(Er)pc T °. At the same time, the number of states encountered within a distance r is proportional only to r (and not r 3) so that, ¢=2r_~
a
1
rTO(kT ) .
(10)
Optimization yields
Ropt =Ro exp(~p,) =Ro e x p ( T o / T ) ' / 2 ,
( 11 )
which leads to a larger conductivity than in eq. (9). Moreover, this result is independent of the dimensionality of the system, and is in agreement with that
Volume 151, number 3,4
PHYSICS LETTERS A
of Efros not only with regard to the exponent, but also with regard to the value of To which is given by eE/~ak. Thus one-dimensional paths define completely the current carrying network in any dimension, and a single electron percolation theory can be justified (further justification is given below). We can also calculate the pre-exponential. It is proportional to the separation of the transitions with the slowest rates, which scales with to, and inversely proportional to the areal density of paths. The areal density of paths is r~ -2 in three dimensions. Thus,
e 2 Pph k T ad¢- k T r, -~g e x p [ - ( T ° / T ) ' / 2 ] =~Uph exp[ -- (To~T) 1/2 ]
(12)
in three dimensions, while e2
ad¢ = ~-~ Uphexp[ - (To~T) 1/2 ]
3 December 1990
One final point regarding transport should be mentioned. Sequential correlations are not expected to modify the above result for the dc conductivity, as all quantities are treated in thermodynamic equilibrium. The low frequency ac conductivity, however, may be strongly affected by sequential correlations, as it is clear that the relaxation times of any non-local relaxation processes will depend strongly on whether the hops are sequentially correlated. A future publication will deal with this topic. An interesting question is: How long does it take, say, for eight domains of size r0 to coalesce to form one domain of size 2ro, when the temperature is dropped by a factor two? If many-electron transitions were required (for reasons of energy) such a transition could require a time proportional to toc exp [ (2ro/rg) 3rg/4a] oc exp [ (Eg/kT)
(14)
(13)
in two dimensions. An important result is that transport takes place along strictly one-dimensional paths, whose density of states if given by one-dimensional results. In l-d the importance of Coulomb interactions can be easily neutralized by disorder, as the total Coulomb energy and the total disorder energy both depend linearly on the concentration of the electrons. In two and three dimensions the Coulomb interaction energy is a, sublinear function of the concentration guaranteeing that the electrons near the Fermi level feel the Coulomb interaction even in the limit of large disorder. This result could explain why the existence of the Coulomb gap has not been conclusively demonstrated through transport measurements. As a further observation we mention that, as no Coulomb barriers are encountered (on account of the degeneracies along the domain boundaries discussed, and on account of the finite density of states at E = E f ) multi-electron transitions need play no role in overcoming such barriers. Moreover, it has been shown [6 ] that correlated many-electron probably plays no role in transport already in two dimensions, so that in one dimension no effect should be seen. As a consequence (in contrast to previous claims by the author [ 7 ] as well as by Pollak and Knotek [ 8 ] ) no particular effect on the dc conductivity due to multielectron transitions is expected.
3rg/4a],
which can be enormous for k T m u c h smaller than Eg. Single-electron transitions could require (at most) a time toc exp (Eg/kT)
( 15 )
each, while the sequential correlation of such transitions could at most extend the time to
toc (Eg/kT) 3 e x p ( E J k T ) ,
(16)
which is obviously much smaller than eq. (14) for Eg >> kT. Thus many-electron transitions appear irrelevant for the approach to thermal equilibrium as well, as result which would explain the relative success of the computer simulations of Davies et al. [ 9 ] and Gruenewald et al. [ 10 ]. The fact, however, that r0 diverges in this limit T-,0, suggests that the DOS obtained by these authors in the limit of zero energy and at zero temperature is unreliable. And in fact, in the limit of zero energy, computer simulations have been unable to distinguish between the quadratic gap and other harder gaps which have been proposed. Moreover the fundamental difference between the system at T = 0 and at any finite temperatures makes questionable any approach to transport based on zero temperature statistics. As a final comment on thermodynamics we point out that the Efros density of states has often been treated as an established fact, but it is strictly speaking an upper limit. It is tempting to argue that the 189
Volume 151, number 3,4
PHYSICS LETTERS A
density o f states given here is a lower limit, as in a real system the " f e r r o m a g n e t i c " o r d e r i n g is n o t expected to be perfect. Since the lower l i m i t a n d the u p p e r l i m i t are in exact agreement, it is felt here that this q u e s t i o n has also b e e n settled once a n d for all.
References
[ 1] A. Hunt, submitted to Philos. Mag. Lett. (1990). [ 2 ] A.L. Efros, J. Phys. C 9 (1976) 2021.
190
3 December 1990
[3] A. Hunt and M. Pollak, J. Phys. C 18 (1985) 5325. [4 ] A.L. Efros, private communication (1990). [5] N.F. Mort, Philos. Mag. 19 (1969) 835. 16] R. Chicon, M. Ortuno, B. Hadley and M. Pollak, Philos. Mag. B 58 (1988) 69. [7] A. Hunt, J. Phys. C 20 (1987) 1469. [8] M. PoUak and M.L. Knotek, J. Non-Cryst. Solids 32 (1979) 141. [9]J. Davies, P.A. Lee and T.M. Rice, Phys. Rev. Lett. 49 (1982) 758. [ 10] M. Gruenewald, B. Pohlmann, L. Schweitzer and D. Wuertz, J. Phys. C 15 (1982) 1153.