hamiltonian tb

7/29/2019 Hamiltonian TB

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2

Tight-binding model

Tight-binding Hamiltonian

Matrix Green functions and self-energies

Semi-infinite leads

Fisher-Lee formula in the GF representation

D.A. Ryndyk

2.1 Tight-binding Hamiltonian

The tight-binding (TB) model was proposed to describe quantum systems inwhich the localized electronic states play essential role, it is widely used asan alternative to the plane wave description of electrons in solids, and alsoas a method to calculate the electronic structure of molecules in quantumchemistry. The main idea of the method is to represent the wave functionof a quantum system (QS) as a linear combination of some known localizedstates (r) (for example, atomic orbitals, in this particular case the methodis called LCAO linear combination of atomic orbitals)

(r) = c(r). (2.1)

Using the Dirak notations | (r) and assuming that (r) are or-thonormal functions | = we can write the tight-binding Hamiltonianin the Hilbert space formed by (r)

H =

(+ e)||+

t||+,

V,||||+ ..., (2.2)

the first term in this Hamiltonian describes the localized states with energies, is the electrical potential, the second term is hopping between the sites,t is the hopping matrix element, which is nonzero as a rule for nearestneighbor cites, the third term is the two-particle interaction, et cetera.

This approach was developed originally as an approximate method, if thewave functions of isolated atoms are taken as a basis wave functions (r),but also can be formulated exactly with the help of Wannier functions. Onlyin the last case the expansion (2.1) and the Hamiltonian (2.2) are exact, butsome extension to the arbitrary basis functions is possible. In principle, theTB model is reasonable only when local states can be orthogonalized. Themethod is useful to calculate the conductance of complex quantum systems


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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 Schrodinger equation, thus it is routinely used in numericalcalculations.

Without interactions we can consider the first two terms of (2.2)

H =

( + e) || +=

t|| (2.3)

as a single-particle Hamiltonian. To solve the single-particle problem it isconvenient to introduce a new representation, where the coefficients c in theexpansion (2.1) are the components of a vector wave function

=

c1c2...

cN

, (2.4)

and are to be found from the matrix Schrodinger equation

H = E , (2.5)

with the matrix elements of the single-particle Hamiltonian

H =

|H

|

= (r)H(r)(r)dr.

1

2 1N N

t t t

Fig. 2.1. A linear chain of sites.


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2.2 Matrix Green functions and self-energies 39

Now let us consider some typical QSs, for which the TB method is appro-

priate starting point. The simplest example is a single quantum dot, the basisis formed by the eigenstates, the corresponding TB Hamiltonian is diagonal

H =

1 0 0 00 2 0 0...

. . .. . .

. . ....

0 0 N1 00 0 0 N

. (2.6)

The next typical example is a linear chain of single-state sites with onlynearest-neighbor couplings (Fig. 2.1)

H =

1 t 0 0t 2 t 0...

. . .. . .

. . ....

0 t N1 t0 0 t N

. (2.7)

The method is applied as well to consider the semi-infinite leads.In the second quantized form the tight-binding Hamiltonian is

H =

( + e) cc +

=

tcc. (2.8)

2.2 Matrix Green functions and self-energies

The solution of single-particle quantum problems, formulated with the helpof a tight-binding Hamiltonian, is possible along the usual line of finding thewave-functions on a lattice, solving the Schrodinger equation (2.5). The othermethod, namely matrix Green functions, considered in this section, was foundto be more convenient for transport calculations, especially when interactionsare included.

The retarded single-particle matrix Green function GR() is determinedby the equation

[( + i)I H] GR = I, (2.9)

where is an infinitesimally small positive number = +0.For an isolated noninteracting system the Green function is simply ob-tained after the matrix inversion

GR = [( + i)I H]1 . (2.10)

Let us consider the trivial example of a two-level system with the Hamiltonian


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H = 1 t

t 2

. (2.11)

The retarded GF is easy found to be

GR() =1

( 1)( 2) + t2

2 tt 1

. (2.12)

Now let us consider the case, when the QS of interest is coupled to twocontacts (Fig. 2.2). We assume here that the contacts are also described bythe tight-binding model and by the matrix GFs. Actually, the semi-infinitecontacts should be described by the matrix of infinite dimension. We shallconsider the semi-infinite contacts in the next section.

Let us present the full Hamiltonian of the considered system in a following

block form

H =

H0L HLS 0HLS H0S HRS

0 HRS H0R

, (2.13)

where H0L, H0S, and H

0R 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 = HLS, HSR = H

RS. (2.14)

The equation (2.9) can be written as

E H0L

HLS 0

HLS E H0S HRS0 HRS E H0R

GLL GLS 0GSL GSS GSR

0 GRS GRR

= I, (2.15)

System

L R

0

HL0

HR0H

S

HLS

HRS

Fig. 2.2. A system coupled to the left and right leads.


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2.3 Semi-infinite leads 41

where we introduce the matrix E = ( + i)I, and represented the matrix

Green function in a convenient form. Now our first goal is to find the systemGreen function GSS which defines all quantities related to the QS only. Fromthe matrix equation (2.15) one has

E H0L

GLS HLSGSS = 0, (2.16)HLSGLS +

E H0S

GSS HRSGRS = I, (2.17)

HRSGSS +

E H0R

GRS = 0. (2.18)

From the first and the third equation

GLS =

E H0L1

HLSGSS, (2.19)

GRS = E H0R1 HRSGSS, (2.20)

and substituting it into the second equation we arrive at the equationE H0S

GSS = I, (2.21)

where we introduce the self-energy

= HLS

E H0L1

HLS+ HRS

E H0R

1HRS. (2.22)

We found, that the retarded GF of a QS coupled to the leads is determinedby the expression

GRSS() =

( + i)I H0S

1

, (2.23)

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

leads G0ii =

E H0R1

(i = L, R)

i = HiS

E H0i

1HiS = H

iSG

0iiHiS, (2.24)

it means, that the self-energy is independent of the state of the QS. Later weshall see that this property conserves also for interacting System, if the leadsare noninteracting.

Finally, we should note, that the Green functions considered in this section,are single-particleGFs, 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). Thedirect matrix multiplication can not be performed in this case. The spectrumof an infinite system is continuous. We should transform the expression (2.24)into some other form.


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0

0

t t

System

Fig. 2.3. A system coupled to a semi-infinite 1D lead.

To proceed, we use the following relation between the lattice Green func-tion and the eigenfunctions () of a system

GR() =

()()

+ i , (2.25)

where is the TB state (site) index, denotes the eigenstate, is the energyof the eigenstate. The summation in this formula can be easy replaced by theintegration in the case of a continuous spectrum. It is important to notice,

that the eigenfunctions () should be calculated for the separately takensemi-infinite lead.

For example, for the semi-infinite 1D chain of single-state sites (n, m =1, 2,...)

GRnm() =

dk

2

k(n)k(m)

+ i k , (2.26)

with the eigenfunctions k(n) =

2sin 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 ofthe self-energy

= V1V1G

R11, (2.27)

where the matrix element V1 describes the coupling between the end site ofthe lead (n = m = 1) and the state | of the System.

To make clear the main physical properties of the lead self-energy, let usanalyze in detail the semi-infinite 1D lead with the Green function (2.26). Theintegral can be calculated analytically (Datta II, p. 213, [12])

GR11() =1

sin2 kdk

+ i 0 2t cos k = exp(iK())

t, (2.28)


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2.3 Semi-infinite leads 43

-2 -1 0 1 2

/t

(

)

Im

Re

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 followingexpressions for the real and imaginary part of the self-energy

Re =|V1|2

t

2 1 [( 1) ( 1)]

, (2.29)

Im = |V1|2

t 1 2(1 ||), (2.30)

= 02t

. (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 wecan make looking at the formulas and the curves.

(i) The self-energy is a complex function, the real part describes the energyshift of the level, and the imaginary part describes broadening. The finiteimaginary part appears as a result of the continuous spectrum in the leads.The broadening is described traditionally by the matrix

= i

. (2.32)(ii) In the wide-band limit (t ), at the energies 0 t, it is possibleto 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

L(R) = iL(R)

2. (2.33)


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2.4 Fisher-Lee formula in the GF representation

After all, we want again to calculate the current through the System. We as-sume, as before, that the contacts are equilibrium, and there is the voltage Vapplied between the left and right contacts. The calculation of the current in ageneral case is more convenient to perform using the full power of the nonequi-librium Green function method. Here we present a simplified approach, validfor noninteracting systems only, following the paper of Paulsson [13].

Let us come back to the Schrodinger equation (2.5) in the matrix repre-sentation, and write it in the following form

H0L HLS 0

HLS H

0S H

RS

0 HRS H0

R

LS

R

= E

LS

R

, (2.34)

where L, S, and R 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 calltrue scattering because we do not define explicitly the geometry of the leads).Namely, in the left lead L =

0L+

rL, where

0L is the incoming wave the

eigenstate of the H0L, and the reflected wave rL, as well as the transmitted

wave in the right lead R, appear only as a result of the interaction betweensubsystems. Solving the equation (2.34) with these conditions, we find

L =

1 + G0LHLSGRH

LS

0L, (2.35)

R = G0RHRSG

R

H

LS0L (2.36)

S = GRH

LS

0L. (2.37)

The physical sense of this expressions is quite transparent, they describe thequantum amplitudes of the scattering processes. Note, that GR here is thefull GF including the lead self-energies.

Now the next step. We want to calculate the current. The partial (at oneenergy) current from the lead to the System is (see the problem 2.5.3)

ji=L,R =ie

h

i HiSS SHiSi

. (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 aresult the Landauer formula is obtained. We present the calculation for thetransmission function. First, after substitution of the wave functions we havefor the partial current going through the system


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2.4 Fisher-Lee formula in the GF representation 45

jR

=ie

h

RHRS

S

SH

RSR =

ie

h

0

L HLSG

AHRS

G

0R G0R

HRSG

RHLS

0L

=

eh

0

L HLSG

ARGRH

LS

0L

. (2.39)

Now we can calculate the transmission function

T = 2

(E E)

0

LHLSGARG

RHLS

0L

= 2

(E E)

0

LHLS

G

ARGRH

LS

0L

=

GARGRHLS

2

(E E)0L0LHLS= Tr

LG

ARGR

. (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 isvery convenient for numerical calculations

T = Tr

tt

= Tr

LGARG

R

. (2.41)

Finally, one important remark, at finite voltage the diagonal energies inthe Hamiltonians H0L, H

0S, and H

0R are shifted

+ e. Consequently,

the energy dependencies of the self-energies defined by (2.24) are also changedand the lead self-energies are voltage dependent. However, it is convenient todefine the self-energies using the Hamiltonians at zero voltage, in that case thevoltage dependence should be explicitly shown in the transmission formula

T() = Tr

L( eL)GR()R( eR)GA()

. (2.42)


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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 andfrom the right to double-site systems

1

1

2

2

1t LV 2tRV

0

2.5.2 Current though a single level in the wide-band limit

Calculate the spectral function, the transmission function, and the conduc-tance 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 leadsin the conductance of a molecular wire, Chem. Phys. 281, 465 (2002).

12. M. Paulsson, Non Equilibrium Greens Functions for Dummies: Intro-

duction to the One Particle NEGF equations, cond-mat/0210519 (2002).

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