2
D.A. Ryndyk
Tight-binding model
Tight-binding Hamiltonian
Matrix Green functions and self-energies
Semi-infinite leads
Fisher-Lee formula in the GF representation
2.1 Tight-binding Hamiltonian
The tight-binding (TB) model was proposed to describe quantum systems in
which the localized electronic states play essential role, it is widely used as
an alternative to the plane wave description of electrons in solids, and also
as a method to calculate the electronic structure of molecules in quantum
chemistry. The main idea of the method is to represent the wave function
of a quantum system (QS) as a linear combination of some known localized
states ψα (r) (for example, atomic orbitals, in this particular case the method
is called LCAO – linear combination of atomic orbitals)
cα ψα (r).
(2.1)
ψ(r) =
α
Using the Dirak notations |α ≡ ψα (r) and assuming that ψα (r) are orthonormal functions α|β = δαβ we can write the tight-binding Hamiltonian
in the Hilbert space formed by ψα (r)
Ĥ =
(α + eϕα )|αα| +
tαβ |αβ| +
Vαβ,δγ |α|βδ|γ| + ..., (2.2)
α
αβ
αβ,δγ
the first term in this Hamiltonian describes the localized states with energies
α , ϕα is the electrical potential, the second term is hopping between the sites,
tαβ is the ”hopping” matrix element, which is nonzero as a rule for nearest
neighbor cites, the third term is the two-particle interaction, et cetera.
This approach was developed originally as an approximate method, if the
wave functions of isolated atoms are taken as a basis wave functions ψα (r),
but also can be formulated exactly with the help of Wannier functions. Only
in the last case the expansion (2.1) and the Hamiltonian (2.2) are exact, but
some extension to the arbitrary basis functions is possible. In principle, the
TB model is reasonable only when local states can be orthogonalized. The
method is useful to calculate the conductance of complex quantum systems
38
2 Tight-binding model.
in combination with ab initio methods. It is particular important to describe
small molecules, when the atomic orbitals form the basis.
In the mathematical sense, the TB model is a discrete (grid) version of
the continuous Schrödinger equation, thus it is routinely used in numerical
calculations.
Without interactions we can consider the first two terms of (2.2)
Ĥ =
(α + eϕα ) |αα| +
tαβ |αβ|
(2.3)
α=β
α
as a single-particle Hamiltonian. To solve the single-particle problem it is
convenient to introduce a new representation, where the coefficients cα in the
expansion (2.1) are the components of a vector wave function
 
c1
 c2 
 
(2.4)
Ψ =  . ,
 .. 
cN
and are to be found from the matrix Schrödinger equation
HΨ = EΨ,
(2.5)
with the matrix elements of the single-particle Hamiltonian
Hαβ = α|Ĥ|β = ψα∗ (r)Ĥ(r)ψβ (r)dr.
ε N −1ε N
ε1 ε 2
t
t
t
Fig. 2.1. A linear chain of sites.
2.2 Matrix Green functions and self-energies
39
Now let us consider some typical QSs, for which the TB method is appropriate starting point. The simplest example is a single quantum dot, the basis
is formed by the eigenstates, the corresponding TB Hamiltonian is diagonal


1 0 0 · · · 0
 0 2 0 · · · 0 




(2.6)
H =  ... . . . . . . . . . ...  .


 0 · · · 0 N −1 0 
0 · · · 0 0 N
The next typical example is a linear
nearest-neighbor couplings (Fig. 2.1)

1 t 0
 t 2 t


H =  ... . . . . . .

 0 ··· t
0 ··· 0
chain of single-state sites with only
···
···
..
.
0
0
..
.
N −1
t




.

t 
N
The method is applied as well to consider the semi-infinite leads.
In the second quantized form the tight-binding Hamiltonian is
Ĥ =
(α + eϕα ) c†α cα +
tαβ c†α cβ .
α
(2.7)
(2.8)
α=β
2.2 Matrix Green functions and self-energies
The solution of single-particle quantum problems, formulated with the help
of a tight-binding Hamiltonian, is possible along the usual line of finding the
wave-functions on a lattice, solving the Schrödinger equation (2.5). The other
method, namely matrix Green functions, considered in this section, was found
to be more convenient for transport calculations, especially when interactions
are included.
The retarded single-particle matrix Green function GR () is determined
by the equation
(2.9)
[( + iη)I − H] GR = I,
where η is an infinitesimally small positive number η = +0.
For an isolated noninteracting system the Green function is simply obtained after the matrix inversion
GR = [( + iη)I − H]−1 .
(2.10)
Let us consider the trivial example of a two-level system with the Hamiltonian
40
2 Tight-binding model.
1 t
t 2
H=
.
(2.11)
The retarded GF is easy found to be
GR () =
1
( − 1 )( − 2 ) + t2
− 2 t
t − 1
.
(2.12)
Now let us consider the case, when the QS of interest is coupled to two
contacts (Fig. 2.2). We assume here that the contacts are also described by
the tight-binding model and by the matrix GFs. Actually, the semi-infinite
contacts should be described by the matrix of infinite dimension. We shall
consider the semi-infinite contacts in the next section.
Let us present the full Hamiltonian of the considered system in a following
block form

 0
HL HLS 0
(2.13)
H =  H†LS H0S H†RS  ,
0
0 HRS HR
where H0L , H0S , and H0R are Hamiltonians of the left lead, the system, and the
right lead separately. And the off-diagonal terms describe QS-to-lead coupling.
The Hamiltonian should be hermitian, so that
HSL = H†LS ,
HSR = H†RS .
(2.14)
The equation (2.9) can be written as



E − H0L −HLS
0
GLL GLS 0
 −H† E − H0 −H†   GSL GSS GSR  = I,
S
LS
RS
0 GRS GRR
0
−HRS E − H0R
System
H LS
H RS
L
H 0L
R
H
0
S
H 0R
Fig. 2.2. A system coupled to the left and right leads.
(2.15)
2.3 Semi-infinite leads
41
where we introduce the matrix E = ( + iη)I, and represented the matrix
Green function in a convenient form. Now our first goal is to find the system
Green function GSS which defines all quantities related to the QS only. From
the matrix equation (2.15) one has
E − H0L GLS − HLS GSS = 0,
−H†LS GLS
GSS − H†RS GRS
+ E−
−HRS GSS + E − H0R GRS
H0S
(2.16)
= I,
(2.17)
= 0.
(2.18)
From the first and the third equation
GLS = E − H0L
GRS = E −
−1
−1
H0R
HLS GSS ,
(2.19)
HRS GSS ,
(2.20)
and substituting it into the second equation we arrive at the equation
E − H0S − Σ GSS = I,
(2.21)
where we introduce the self-energy
Σ = H†LS E − H0L
−1
HLS + H†RS E − H0R
−1
HRS .
(2.22)
We found, that the retarded GF of a QS coupled to the leads is determined
by the expression
−1
0
GR
,
(2.23)
SS () = ( + iη)I − HS − Σ
the effects of the leads are included through the self-energy.
Here we should stress the important property of the self-energy (2.22), it
is determined only by the coupling Hamiltonians, and the GFs of the isolated
−1
leads G0ii = E − H0R
(i = L, R)
Σi = H†iS E − H0i
−1
HiS = H†iS G0ii HiS ,
(2.24)
it means, that the self-energy is independent of the state of the QS. Later we
shall see that this property conserves also for interacting System, if the leads
are noninteracting.
Finally, we should note, that the Green functions considered in this section,
are single-particle GFs, and can be used only for noninteracting systems.
2.3 Semi-infinite leads
Let us consider now a system coupled to a semi-infinite lead (Fig. 2.3). The
direct matrix multiplication can not be performed in this case. The spectrum
of an infinite system is continuous. We should transform the expression (2.24)
into some other form.
42
2 Tight-binding model.
System
ε0 ε0
t
t
Fig. 2.3. A system coupled to a semi-infinite 1D lead.
To proceed, we use the following relation between the lattice Green function and the eigenfunctions ψλ (α) of a system
GR
αβ () =
ψλ (α)ψ ∗ (β)
λ
λ
+ iη − λ
,
(2.25)
where α is the TB state (site) index, λ denotes the eigenstate, λ is the energy
of the eigenstate. The summation in this formula can be easy replaced by the
integration in the case of a continuous spectrum. It is important to notice,
that the eigenfunctions ψλ (α) should be calculated for the separately taken
semi-infinite lead.
For example, for the semi-infinite 1D chain of single-state sites (n, m =
1, 2, ...)
π
dk ψk (n)ψk∗ (m)
GR
()
=
,
(2.26)
nm
−π 2π + iη − k
√
with the eigenfunctions ψk (n) = 2 sin kn, k = 0 + 2t cos k.
Let us consider a simple situation, when the QS is coupled only to the end
site of the 1D lead (Fig. 2.3). From (2.24) we obtain the matrix elements of
the self-energy
∗
V1β GR
(2.27)
Σαβ = V1α
11 ,
where the matrix element V1α describes the coupling between the end site of
the lead (n = m = 1) and the state |α of the System.
To make clear the main physical properties of the lead self-energy, let us
analyze in detail the semi-infinite 1D lead with the Green function (2.26). The
integral can be calculated analytically (Datta II, p. 213, [12])
exp(iK())
sin2 kdk
1 π
R
=−
,
(2.28)
G11 () =
π −π + iη − 0 − 2t cos k
t
43
Σ(ε)
2.3 Semi-infinite leads
Im Σ
Re Σ
-2
-1
0
ε/t
1
2
Fig. 2.4. Real and imaginary parts of the contact self-energy as a function of energy.
K() is determined from = 0 + 2t cos K. Finally, we obtain the following
expressions for the real and imaginary part of the self-energy
ReΣαα =
|V1α |2 κ − κ2 − 1 [θ(κ − 1) − θ(−κ − 1)] ,
t
|V1α |2 ImΣαα = −
1 − κ2 θ(1 − |κ|),
t
− 0
.
κ=
2t
(2.29)
(2.30)
(2.31)
The real and imaginary parts of the self-energy, given by these expressions,
are shown in Fig. 2.4. There are several important general conclusion that we
can make looking at the formulas and the curves.
(i) The self-energy is a complex function, the real part describes the energy
shift of the level, and the imaginary part describes broadening. The finite
imaginary part appears as a result of the continuous spectrum in the leads.
The broadening is described traditionally by the matrix
Γ = i Σ − Σ† .
(2.32)
(ii) In the wide-band limit (t → ∞), at the energies −0 t, it is possible
to neglect the real part of the self-energy, and the only effect of the leads is
level broadening. So that the self-energy of the left (right) lead is
ΣL(R) = −i
ΓL(R)
.
2
(2.33)
44
2 Tight-binding model.
2.4 Fisher-Lee formula in the GF representation
After all, we want again to calculate the current through the System. We assume, as before, that the contacts are equilibrium, and there is the voltage V
applied between the left and right contacts. The calculation of the current in a
general case is more convenient to perform using the full power of the nonequilibrium Green function method. Here we present a simplified approach, valid
for noninteracting systems only, following the paper of Paulsson [13].
Let us come back to the Schrödinger equation (2.5) in the matrix representation, and write it in the following form




 0
HL HLS 0
ΨL
ΨL
 H† H0 H†   ΨS  = E  ΨS  ,
(2.34)
S
LS
RS
0
Ψ
Ψ
0 HRS HR
R
R
where ΨL , ΨS , and ΨR are vector wave functions of the left lead, the System,
and the right lead correspondingly.
Now we find the solution in the ”scattering form” (which is difficult to call
true scattering because we do not define explicitly the geometry of the leads).
Namely, in the left lead ΨL = ΨL0 + ΨLr , where ΨL0 is the ”incoming wave” – the
eigenstate of the H0L , and the ”reflected wave” ΨLr , as well as the ”transmitted
wave” in the right lead ΨR , appear only as a result of the interaction between
subsystems. Solving the equation (2.34) with these conditions, we find
ΨL = 1 + G0L HLS GR H†LS ΨL0 ,
(2.35)
ΨR = G0R HRS GR H†LS ΨL0
R
ΨS = G
H†LS ΨL0 .
(2.36)
(2.37)
The physical sense of this expressions is quite transparent, they describe the
quantum amplitudes of the scattering processes. Note, that GR here is the
full GF including the lead self-energies.
Now the next step. We want to calculate the current. The partial (at one
energy) current from the lead to the System is (see the problem 2.5.3)
ji=L,R =
ie †
Ψi HiS ΨS − ΨS† H†iS Ψi .
h̄
(2.38)
To calculate the total current we should substitute the expressions for
the wave functions (2.35)-(2.37), and summarize all contributions [13]. As a
result the Landauer formula is obtained. We present the calculation for the
transmission function. First, after substitution of the wave functions we have
for the partial current going through the system
2.4 Fisher-Lee formula in the GF representation
ie †
jR =
ΨR HRS ΨS − ΨS† H†RS ΨR =
h̄
ie 0†
0
R †
0
H
ΨL HLS GA H†RS G0†
−
G
G
H
Ψ
RS
R
R
LS L =
h̄ e
Ψ 0† HLS GA ΓR GR H†LS ΨL0 .
−
h̄ L
45
(2.39)
Now we can calculate the transmission function
0†
0
T = 2π
δ(E − Eλ ) ΨLλ
HLS GA ΓR GR H†LS ΨLλ
= 2π
=
δ
λ
λ
0†
0
δ(E − Eλ ) ΨLλ
HLS Ψδ Ψδ† GA ΓR GR H†LS ΨLλ
δ
Ψδ† GA ΓR GR H†LS
2π
δ(E −
0†
0
Eλ )ΨLλ
ΨLλ
HLS Ψδ
λ
= Tr ΓL GA ΓR GR .
(2.40)
To evaluate the sum in brackets we used the eigenfunction expansion (2.25)
for the left contact.
We obtained the new representation for the Fisher-Lee formula, which is
very convenient for numerical calculations
T = Tr t̂t̂† = Tr ΓL GA ΓR GR .
(2.41)
Finally, one important remark, at finite voltage the diagonal energies in
the Hamiltonians H0L , H0S , and H0R are shifted α → α + eϕα . Consequently,
the energy dependencies of the self-energies defined by (2.24) are also changed
and the lead self-energies are voltage dependent. However, it is convenient to
define the self-energies using the Hamiltonians at zero voltage, in that case the
voltage dependence should be explicitly shown in the transmission formula
T () = Tr ΓL ( − eϕL )GR ()ΓR ( − eϕR )GA () .
(2.42)
46
2 Tight-binding model.
2.5 Problems
2.5.1 Toy example of the self-energy
Calculate the self-energy for a single-level system, coupled from the left and
from the right to double-site systems
ε1 ε1
t1
ε0
VL
ε2 ε2
VR
t2
2.5.2 Current though a single level in the wide-band limit
Calculate the spectral function, the transmission function, and the conductance though a single level in the wide-band limit
2.5.3 Current in the TB method
Prove the formula (2.38).
Additional reading
• D.K. Ferry and S.M. Goodnick, ”Transport in nanostructures”, sect. 3.8.
• S. Datta, ”Quantum transport: atom to transistor”, chapter 8.
• S. Datta, ”Electronic transport in mesoscopic systems”, sections 3.5, 3.6.
Bibliography
11. G. Cuniberti, G. Fagas, and K. Richter, “Fingerprints of mesoscopic leads
in the conductance of a molecular wire,” Chem. Phys. 281, 465 (2002).
12. M. Paulsson, “Non Equilibrium Green’s Functions for Dummies: Introduction to the One Particle NEGF equations,” cond-mat/0210519 (2002).