ХИМИЧЕСКАЯ ФИЗИКА, 2004, том 23, № 2, с. 128-139
ЭЛЕМЕНТАРНЫЕ ^^^^^^^^
ФИЗИКО-ХИМИЧЕСКИЕ ПРОЦЕССЫ
УДК 539.194
MULTI-CENTER SCATTERING THEORY OF ELECTRON TUNNELING TRANSITIONS
© 2004 г. M. A. Kozhushner, R. R. Muryasov, V. S. Posvyanskii
N.N. Semenov Institute of Chemical Physics, Moscow, Russia Received 16.11.2002
The theory of the "bridge" effect, i.e. electron tunneling with the participation of intermediate particles, is developed. It is shown and proved the general incorrectness of the approach that operates with the restricted basis of the bound states on the bridge particles to obtain sub-barrier electron Green function. This approach is adequate only in the energy region near the bound energy of the bridge centers at the large center-to-center distances. Our theory is based on the concept of multiple sub-barrier scattering of tunneling electron on the intermediate particles. The regular method of the calculation of the energies of the collectivized bound states as the poles of the total scattering amplitude is developed. It is shown that probability of the tunneling transition depends on the value and sign of the amplitude of electron sub-barrier scattering on one particle, and this amplitude can be calculated by the variation-asymptotic method developed by the authors earlier. The bridge induced enhancement of the tunneling probability depends exponentially on the length of the bridge, the exponent rises with modulus of the tunneling energy, i.e. with energy decreasing. Such enhancement is possible also in the absence of the bound states on the bridge. Two modes of the tunneling transitions, adiabatic and non-adiabatic ones, are considered.
1. INTRODUCTION
The electron tunnel transition is the basis of many important physical processes such as field emission [1], hopping in doped semiconductors [2], donor-acceptor transition of photo-excited electrons in crystals and glasses [3, 4] and the electron transition in biological systems [5-7]. In addition, the electron tunnel transition is a tunnel current in scanning tunneling microscopy (STM) [8, 9] or in one-molecule electronic devices [10, 11]. Very frequently some particles in the barrier region influence electron tunneling. In a condensed matter where the donor-acceptor pairs are arranged, they are the intermediate particles that break the homogeneity between a donor and an acceptor. In biological systems they are the intermediate fragments of proteins, which significantly change the probability of tunneling transition. This influence is the essence of so-called "bridge effect" [12-15]. The DNA oligomer conductivity that was measured in a recent brilliant experiment [16] can be represented as a resonance tunneling through the collective levels of an electron on the oligomer.
A considerable number of works was devoted to the theory of intermediate particles influence on tunneling of electrons [4, 14, 17-30]. The theoretical methods in these references can be classified in two groups. The first ones are applied to the conductivity in STM and molecular wires. They are based on the calculation of the wave function for the full system by using some well-developed quantum-chemical methods such as the most frequently used method of local density approximation (LDA). According to this method the current at different biases can be calculated on the base of the obtained
wave function. However, all the quantum-chemical methods, including LDA, do not work accurately enough when the density of conducting electron (with the energy close to Fermi level) is sufficiently small on the intermediate molecule. It means that this molecule is inside the asymptotic region of the tunneling electron's wave function [31]. The simultaneous calculation of the values of the wave function in the main region of electron localization and in the remote asymptotic region by some quantum-chemical method seems to be impossible, particularly, for the complicated many-electron systems. In this case the influence of the intermediate particle on tunneling transition amplitude, and consequently on the current, can be described as electron sub-barrier scattering on this particle. The variation-asymptotic method was developed in [32, 33] for calculating the sub-barrier scattering amplitude and, therefore, for calculating the tunneling current in STM and other tunneling transitions.
The methods of second group that are applied to calculate the bridge effect in an donor-acceptor pair have, schematically, the following structure. The electron tunneling probability WET is given by the golden rule expression (the atomic units are used all over the article):
Wet = 2 n| Ada| 2 BFC (1)
where ADA is the electronic amplitude of the tunneling transition and BFC is the combined Frank-Condon factor for transition in nuclear modes. BFC defines the temperature dependence of WET. The electron bridge part of the amplitude ADA is expressed as
Fig. 1. The schematic space-potential structure of a donor-acceptor pair with the bridge of 5 particles. The both potential curves and energy levels of the donor and acceptor are pictured with solid lines, dotted lines designate the potentials of the intermediate particles and the energies of the bound states on them. The regions where Uj (r) and Ua (r) are distinct from zero are designed as chain lines (see Section 2).
A
DA
- S v
DiGikVkA•
(2)
i, k
The summation is taken over all orbitals of the bridge; VDi and VkA are the couplings of the electronic states on D (donor) and A (acceptor) with the bridge orbitals; Gik(Et) are the matrix elements of one-electron Green's
function (GF) of bridge Hamiltonian H at a tunneling energy Et,
G( Et) = ( Et - H
(3)
The crucial point of all these methods is using of the restricted basis of the bound states wave functions localized on each particle of the bridge to calculate Gik(Et) and ADA. If only one bound state on each bridge particle, i.e. the negative ion state, is considered, the indexes in the sum in the Eq. (2) denote really the different bridge particles. The space-energy scheme of this case is represented on Fig. 1. We will prove such approach of restricted basis (ARB) to be incorrect in general case.
Let us examine the bridge that includes only one particle. Then, according to ARB
G( r, r'; Et) =
Vi ( r )Vi ( r' ) Et - ei •
(4)
ada -
VDi V1A Et - e 1 •
(5)
What dependence of Ada on the donor-acceptor dis -tance ^DA results from this equation: The interaction VD1 is defined by the expression
V
D1
- Jdr^D(r - Rd)U(r;RD, Ri)y,(r - Ri) (6)
where yD(r - RD) is the wave function of the donor bound state, U(r; RD, R1) is the electron potential energy in the field of centers D and 1. According to asymptotic properties of the wave functions [31, 34],
¥d ( r - Rd ) ¥i( r - R i )
lr - R,
- exp(-x|Ri - Rd| ), exp(-x, |R, - RD )
(7)
where x = (2|EJ)1/2 (the bound energy on the donor is Et) and x1 = (2|e1|)1/2. For tunneling transitions 0 > £1 > Et, and consequently, x > x1. Then, it results from (6):
V
Di
exp(-x, |ri - rd )
(8)
Just as in (8), W1A ~ exp(-x1|RA - R1|). If the centers D, A, and 1 are situated approximately on one straight line, we have
Ada ~ exp(-x, |Ra- Re
(9)
This result denotes that the influence of one intermediate particle depends on ^DA exponentially, and if e1 —► 0, the tunneling probability stops to depend exponentially on ^DA. Intuitively, it is clear that such result is incorrect, and it will be shown below the correct result for one intermediate particle is
Ada ~ exp (-x IRa- Rd| ),
(10)
Here, y1(r) is the wave function of the bound state and £1 is its energy. Then, the resulting expression for Ada is
i.e. the exponent is the same as without any intermediate particle, x = (2|£i|)1/2. The intermediate center influences the pre-exponential factor only, and this influence does not really depend on ^DA. What is the source of the improper result (9)?
The general mistake is using of the restricted basis of the bound states wave functions for the determination of GF that gives, particularly, the expression (4) for the one-particle bridge. If the interaction of the intermediate particle with a tunneling electron may be afforded as a potential U1, then GF may be expressed by
the usual spectral representation that is the analogue of the determination (3):
Gv(r, r; £) =
£ £n
dk
¥k( r )y * ( r' -£ - k2 / 2 '
(11)
Here yn are the wave functions of the bound states in the potential U1 with energies £n, and yk are the scattering eigen-functions, yk(r) ~ exp('kr) + exp('kr)a(£, -)/r. The value a(£, -) is the electron scattering amplitude on the potential U1, and a(£, -) has the poles, as the analytic function of k, at the points k = (2e„)1/2 = ±?'(2|£n|)1/2. The integral in Eq. (11) is defined by the residues in a complex plane k. As it is shown in [35-37], at the conditions |r - R1|, |r - R1| > r1 where r1 is the range of the potential U1(r - R1), the contribution of the residue in the pole of the amplitude a(i , - ), k2/2 = i n, is exactly mutually canceled with the contribution of n-th bound state in the sum on the bound states in Ea. (11), for any potential U1(r). As a result of this cancellation, Green's function Gy(r, r'; £) is equal only to the contri -bution of the residue of the pole k = (2i)1/2 in the inte -gral in Eq. (11). The final exact expression for GF that can be derived from Eq. (11) using analytical properties of ^-matrix is
G(r, r'; Et) = Go (r, r'; Et) + + Go( r, Ri; Et) t (Et,-) Go (Ri, r'; Et)
where
Go (r, r'; Et)
= exp(-5-|r - r'| ) = 2 n - r - r' -
(12)
(13)
is GF of a free electron, x = (2|Et|)1/2.
The term t(£, -) is the scattering operator (SO), t(£, -) = = -2na(£, -). When £ < 0 a(£, -) is the analytic continuation of the amplitude at real moments, and - is the angle between two vectors, R1 - r and r' - R1. The ex pression (12) was obtained in [4] using the perturbation theory on U1. Considering of both the electronic struc ture
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