o — o — o — o — o — c > 0 phonons SiteO) Electron motion
Figure 2.6: Schematic diagram showing model of mapping of many body system to one body system. Various energy channels shown, some of which can only be accessed by phonon scattering. For simplicity we only show the channels associated with one phonon mode.
[21, 157]. As an atomic wire is effectively one-dimensional, the Hamiltonian can be
written as one-dimensional and so is dependent on one coordinate only, j . For a one
dimensional wire we are only concerned with the nearest neighbours, so we can replace an arbitrary index K by & term j + I and j — I. The interaction with the harmonic modes should be localised in the atomic wire. The Hamiltonian is thus generally (in eigenvector-space n)
(2.95)
q , n , m
where a+ and a , {c^ and Cn) are the creation and annihilation terms of the phonon mode
q with frequency Ug (of the electron in state n with eigenenergy €„) and 7g„m represents
the coupling of phonon mode q from state n to state m. Nonlinear phonon modes or
nonlinear phonon-phonon coupling can be added to the Hamiltonian by adding terms of
higher than quadratic order, such as (o*^)^ + H.c.. Similarly nonlinear electron-phonon
coupling can be added by adding terms such as c^c{a'^)^ + H.c. (by nonlinear we mean
th at there are terms which are not proportional to the quadratic).
We will generally use the following setup, th at the atomic scale wire (containing Na
one-dimensional metallic leads (labelled L for left and R for right) whose Hamiltonians are +00 . . Hk = ^ ^Kd]dj 4- Pk [ d ] d j - i -t- (2.96) J=JVa + l and Hl = ^ Chd^dj + I3l (d^jdj-i (2.97) j=—oo
via the coupling matrices
Tr — Ï/R + ^\jadNa + l^ (2.98)
and
Tl = VL (d l c i -f cjdo) • (2.99)
We define the operators d] (dj) to create (annihilate) an electron on a site j inside the
leads, with on-site energy c l.r and nearest neighbour hopping integral /?l,r. The hopping
m atrix elements between the ends of the molecule and the right and left leads are denoted by v l,r. Inside the atomic scale wire it is easy to transform from a site representation
to an eigenstate representation by the operation where Zn{j) are the
components of the nth eigenstate on site j .
We perform the procedure to map the many-body problem onto a single-body problem with many channels by writing the total scattering wavefunction |^ ( ^ ) ) for the total
energy E of the system as
= E
(2-100)
and the basis set used to expand the scattering waves is
i j , K ) ) = 4 n ^ ^ i “ > (2.101)
q V ^ 9 -
where |0) is the ground state of the neutral molecule and (n,) is the set of phonon occupation (how many phonons are occupying each phonon mode). This means th at we have not explicitly separated the electron and phonon degrees of freedom. For hole (or electron) propagation through the atomic scale wire each channel has a different set of phonon occupation numbers (n^). To find the total wavefunction |$(F7)) we have to solve the Schrodinger equation with the total Hamiltonian
H \ ^ { E ) ) = {Hw + Tl + Hl+ Tr + Hr) |^ (E )) = E \ ^{ E ) ) . (2.102)
The total energy E of the system (including electron injection energy) is conserved during
the scattering process (this is only true for dissipationless systems)
E = E \ n + ^ T lq h u !q = E o u t + ^ T U q h u Jq (2.103)
9 9
where Ein and (n^) axe the initial energy and set of phonon occupation numbers and
Eout and (m ,) are the final energy and set of phonon occupation numbers. Only if the process is elastic will (n,) = (m^). We do not assume th at this is the case.
The form of the wavefunction coefficients inside the leads is asymptotic, and corre sponds to Bloch waves propagating inside the leads whose amplitudes are the reflection and transmission coefficients (r(,n,) and t^mg) respectively) of the electron in different channels. For the left lead (where we inject an electron) the coefficients are (for j < —1)
" j .K ) = + r-(m,)e (2.104)
(2.105)
The are the wavevectors of the Bloch waves in different channels. To find the
dispersion relations for these wavevectors we have to solve the Schrodinger equation
(j\H\'9{E)) = E{j\'^{E)). We find th at for the incoming wave the wavevectors are
^ i n = Cl + 2 ^lc o s , (2.106)
for the outgoing reflected wave the wavevectors are
E o n t = € L + 2 / 9l c o s (2.107)
and for the transm itted wave the wavevectors are
- E o u t = C R + 2 / ? r COS . (2.108)
All our transport calculations take place in the limit of low temperature. This means th at we can take the initial set of phonon occupation numbers (n^)=0. If the injected electron’s energy lies outside the channels defined above, th at channel is not propagating and an electron has to tunnel through the gap between leads without any propagation through the wire.
We can relate the reflection or transmission coefficients to the wavefunction coefficients by solving {j \H\^{E)) = E {j \ ^ { E ) ) at site j = 0 and j = Na + 1. The relations would be
^(m,) - (2.110)
where (0) is the elastic channel and (m ,) 7^ 0 are the inelastic channels.
Finally one needs to find the wavefunctions on sites j = 1 . . .Na. As we have de fined the wavefunctions in terms of a series of channels and as we are allowing electrons to propagate only via nearest neighbour sites, finding the wavefunctions thus becomes
equivalent to solving the complex linear system of the form M -g = where g are the
otj,{riq) and d is imaginary. The m atrix M is by definition sparse, but if there is a large number of phonon modes with a lot of different occupations solving this linear system becomes unfeasible. One method of avoiding this numerical problem is to use the prop agation m atrix method (where we can use simpler tight-binding m atrix equations), but as explained above, this can be rewritten as a Green’s function method.
Using a Green’s function method allows us to remove the leads by introducing em bedding potentials. To then find the solution of the full scattering problem we want to propagate a source term |s(F?)) (which is an injected plane wave at energy E) through the wire
|a(£)> = G,ff{£)|s{£)> (2,111)
where
|a ( S ) ) = a„,(„,,(B )|n .(n ,)> (2.112)
n , ( n , )
is the wavefunction of the scattering state and
Geff(£) = [ E - H w - E ^ (£ ) - E '‘(B )]-‘ (2.113)
is the effective Green’s function of the wire connected to the leads. H w is the Hamiltonian
of the two electrodes. These embedding potentials are diagonal matrices in the |n, (n^)) basis set with components
(2.114)
and
(2.115)
where pf.R is the ‘surface’ Green’s function of the isolated leads (see also (2.94)). This is analytically given by g}%j(^) = exp(*A;^^(E))//)L,R.
We also need a form for the source term |s (£?)). This has components given by
(2.116)
E — S(J5) Qfm 2 (E ) a{E) s(E)
T>{E) —E + Hyi/ + %e 2 (E ) Qm a{E) — s( E)
We can simplify the solution of the system (2.113) by explicitly separating the real and imaginary parts of the vectors and matrices as follows
(2,117)
where S = + S ^. This renders the matrix above real and symmetric and thus
amenable to standard computational methods of solution (such as the conjugate gradients technique).
To find |a(E )) we solve G ^ |a ) = |s). The asymptotic outgoing waves on the right lead can be written as
of the wire connected to the right lead. We can relate these coefficients to the inelastic transmission amplitude t { E , E ‘) where E ‘ is the energy of an electron in an outgoing channel after it has been inelastically scattered. This is given by
* (£,£;') = (2.119)
(m ,) q
The effective total transmission probability is then
T ( E ) = Y . (2.120)
( m , ) / l L S m » ,0)
To find the current which is leaving through the outgoing channels in the right electrode, (or reflected from the incoming channels in the left electrode) we use
(2.121)
As we are only considering hopping between adjacent sites, the current in each channel of the right electrode is found to be proportional to the transmission amplitudes.
(2.122)
We have now discussed how we can find the structure of candidate atomic scale wires, and then having found the structure and obtained various pieces of physical information, how we can find the conductivity characteristics of these atomic scale wires, including electron-phonon effects. However we have not yet explained how to obtain such physical
information such as the electron-phonon coupling 7^nm or the phonon spectra. An illus