Dan Wanegar

Nonequilibrium Quantum Mechanics of a Single-Molecule

Heterojunction

Supervisor: Charles Stafford

The ProblemFinding the electron transport through a single-molecule junction.

•Electrons, as fermions, obey the Pauli Exclusion Principle, so learning creation and annihilation operators was necessary.

•The Green’s function, G, which describes the quantum dynamics, was the next step.

•Physics observables, such as electric current, can be expressed in terms of G

The HamiltonianH=Hmol+Hleads+HT

d and d+ are creation and annihilation operators obeying:{di,dj

+}=didj+

+dj+di=i,j

{di,dj}={di+,dj

+}=0

(Hmol)ij = hijdi+dj

(Hleads)k = kck+ck

(HT)nk=Vn,kdn+ck+Vn,k

*ck+dn

The Retarded Green’s Function

Plugging in previous equations followed by a Fourier transform (right) gives

G = [IE – Hmol – ]-1, where

i,j =

The i0+ ensures a retarded function as opposed to advanced

In the broad-band limit, = i(r – a) =And r = a

* is taken to be imaginary, so r = -i/2

Current and The Transmission Matrix

The probability of transmission from lead i to lead j is given by the i,j component of the transmission matrix.

The current into lead i is given by

The transmission matrix is given byTij = Tr(iGrjGa)

A proof can be found in Electronic Transport in Mesoscopic Systems by Supriyo Datta

The General Diatomic Conductor

When 1, 2, and G are entirely arbitrary, the matrix multiplication gives:T12=ijkm

ijkmGjkGim

*

Which can be expanded and simplified due to symmetries to:

T12 = ij |ii||

jj||Grij|+2Re(

1112Gr11

*Gr12 + 12

11Gr11*Gr12

+ 12

12Gr12*Gr21 +

1221Gr11

*Gr22 +

1222Gr12

*Gr22 + 22

12Gr22*Gr21)

In general, the 12 and

12 terms are small enough and complex, so the nasty part usually averages to zero.

The Vertical Diatomic Conductor:Fano Antiresonances

An interesting case to note is the

1.1 and 1.1 being the nonzero in the

connection terms. This gives

G-1=

Since 111 and 2

11 are the only nonzero terms, this means that

T11 = 1 2|G11|2,

Which has exactly one node at E=2.

This can also be looked at as an atomic conductor (atom 1) with three leads (atom 2 being the 2nd). This makes

G =

The Horizontal Diatomic Conductor:Exceptional Points

For a horizontal conductor, the time-dependant transmission function is

[1]

where

So if 1 = 2,

Which is either purely imaginary or purely real. As can be seen above, there is a very large difference between 2V>|’| and 2V<|’|, so |’|=2V is called the exceptional point.

So how does this relate to the energy-dependant transmission function?

[1] D.M.Cardamone, C.A.Stafford, and B.R.Barrett, Phys. Stat. Sol. (b) 230, No. 2 (2002)

What About The Energy-Dependant Transmission Function ?Using the energy-dependant Green’s function from earlier,T12=1

11211|G12|2,

and

Which is especially interesting to look at when 1=2, as it turns into

E =

A Less Distinguishable ResultFor large enough real E and small enough , the transmission function has two peaks. Otherwise, only one peak is observed.

However, E ± lies in the complex plane, but E can only be real. This means that peaks don’t occur at exactly E = Re(E ±).

So for small enough E and large enough , the transmission function doesn’t appear to be qualitatively different for E real or imaginary.

E / = 2.3 E / = 1.2

E / = .59 E / = .22

Taking the n-th Derivative…Shows that even for very small real E, there is still a qualitative difference from imaginary E.

n=0

E / = 2.3 1.2 .59 .22

1

2

5

Conclusions•The nonequilibrium quantum dynamics of electrons in a single-moleule junction can be solved exactly using Green’s functions when neglecting electron-electron interactions•For a diatomic molecule, both a Fano antiresonance and an exceptional point can occur, depending on the geometry.