3. Chapter 1
Introduction
Electron transport theory is a wide ranging field and is essential in understand-
ing processes to do with tunneling and interference effects, taking place on the
mesoscopic scale. Devices like the scanning tunneling microscope are able to
attain atomic resolutions by the quantum mechanical effect of tunneling. This
represents a great motivation to explore the connection between the mechanism
of conduction on mesoscopic scales and its implications for tunneling through a
vacuum gap or a molecule.
The theoretical approach by Landauer provides the framework of the inves-
tigation. In this point of view transport is based on the probability that an
electron will be transmitted through a conductor. The approach is quantum
mechanical and not semi classical as individual electrons are confined to move
in single channels. Transport becomes ballistic instead of diffusive as scattering
from phonons and impurities becomes negligible on mesoscopic scales. An elec-
tric field acting as a force carrier becomes less relevant. Instead the emphasis is
shifted towards the Fermi distribution functions of the contacts of the channel,
affecting electrons to travel through the conductor.
Transmission through the channel becomes probabilistic and the asymmetric
potential barrier is thus investigated as it provides a simple model of transport
through the channel. As a next step a linear chain of atoms representing a
one dimensional conductor is treated as a tight binding wire. In tight binding
theory the crystal Hamiltonian consists of the atomic Hamiltonian and a correc-
tion term which produces a periodic potential with the periodicity of the Bravais
lattice. Eigenstates of the tight binding Hamiltonian are superpositions of the
atomic orbitals. The constitution of the wire is then varied for three cases. In
the simplest case all atoms have equal onsite energies as well as equal hopping
integrals betwen them. Then a break is introduced in the wire and its implica-
tions are investigated. Finally a defect is introduced in the wire containing a
different onsite energy than the rest of the atoms in the chain. The nanowire
is studied as it provides a good understanding of the case of electron trans-
mission through an atom or molecule. Overlapping atomic orbitals imply that
electrons have a non zero velocity throughout the atomic chain. The hopping
integrals imply therefore electron transport via tunneling between neighbouring
atomic sites where the onsite energies of the atoms represent the potential bar-
3
4. rier height. Effectively this is equivalent to the asymmetric potential model. In
theory the transmission of particles through a potential barrier serves as a good
understanding of the one dimensional case of tunneling through a vacuum gap.
The remarkable application of this model is the scanning tunneling microscope
where tunneling through a vacuum gap is its central mechanism. In the more
extensive approach by Tersoff and Hamann this is treated three dimensionally.
By treating it one dimensionally the model of the tight binding wire can be ap-
plied to understand the tunneling process between tip and surface. This model
leaves open the possibility of an adsorbed molecule on the surface. The study
of adsorbate molecules under STM operation reveals their electronic stucture
as transmission resonance is observed. In the case of adsorbate vibrations con-
duction peaks are observed corresponding to correlations between vibrational
modes and transmission. Possibly these are the results of contributing inelastic
channels in the conduction process. The interpretation is referred from Mingo
et al. who consider the probabiliy of excitation of vibrational modes under STM
operation as related to the lifetime of the vibration due to electron hole pair
excitations.
4
5. Chapter 2
Asymmetric potential
barrier
Electrons incident on the left are free particles. The incident electron is thus a
plane wave travelling in the positive x direction. At x = 0 there is the probability
that the particle is reflected back which results in a plane wave of amplitude
R (reflection coefficient) travelling backwards. The wave function in the region
x < 0 is thus a superposition of two plane waves travelling in the positive and
negative direction. In the potential barrier region 0 < x < w the particle is in a
confined space resulting in a standing wave. On the right side of the potential
barrier x > w the electron is a free particle represented by a plane wave with
amplitude T travelling in the positive x direction. The squared reflection and
transmission coefficient give the probabilities of the particle being reflected or
transmitted through the potential barrier.
ϕ (x) = eikx
+ Re−ikx
x < 0 (2.1)
ϕ (x) = Aeikx
+ Be−ikx
0 < x < w (2.2)
ϕ (x) = Teikx
x > w (2.3)
The boundary conditions are that the wave functions and their derivatives
must be piecewise continuous over region one, two and three. Applying these
at the zone boundaries x = 0 and x = w yields a set of simultaneous equations
(2.4) and (2.5), from which coefficients A and B are eliminated. The matrix
equation (2.6) is obtained with the vector containing coefficients R and T which
is inverted, yielding direct expressions for the coefficients R and T only in terms
of the wave vectors.
ikL (1 − R) = kB (A + B) (2.4)
AekBw
+ Be−kBw
= TeikBw
(2.5)
5
6. Figure 2.1: Asymmetric potential barrier
kB + ikL
kB − ikL
=
− (kB − ikL) e−kBw
eikRw
(kB + ikR)
− (kB + ikL) ekBw
eikRw
(kB − ikR)
R
T
(2.6)
R
T
=
1
∆
e−kBw
eikL
(kB − ikR) −e−kBw
eikRw
(kB + ikR)
(kB + ikL) − (kB − ikL)
kB + ikL
kB − ikL
(2.7)
∆ =
− (kB − ikL) e−kBw
eikRw
(kB + ikR)
− (kB + ikL) ekBw
eikRw
(kB − ikR)
(2.8)
∆ = 2eikRw
kRkL − k2
B sinh (kBw) + ikB (kL + kR) cosh (kBw) (2.9)
R =
1
∆
ekBw
eikRw
(kB − ikR) (kB + ikL) − e−kBw
eikRw
(kB + ikR) (kB − ikL)
(2.10)
R =
2eikRw
∆
k2
B + kLkR sinh (kBw) + i (kLkB − kRkB) cosh (kBw) (2.11)
T =
1
∆
(kB + ikL)
2
− (kB − ikL)
2
=
4ikBkL
∆
(2.12)
The reflection and transmission probability can be fully determined by squar-
ing their modulus.
|R|2
=
4
|∆|2
k2
B + kLkR
2
sinh2
(kBw) + (kLkB − kRkB)
2
cosh2
(kBw)
(2.13)
6
7. |T|2
=
16
|∆|2
(kBkL)
2
(2.14)
|R|2
+
kR
kL
|T|2
= 4
|∆|2 4k2
BkRkL + k2
B − kRkL
2
sinh2
(kBw) + (kBkL + kBkR)
2
cosh2
(kBw)
+4k2
BkLkR sinh2
(kBw) − 4k2
BkRkL cosh2
(kBw) (2.15)
kR
kL
|T|2
+ |R|2
= 1 (2.16)
Equation (2.16) is the central result of this calculation and expresses current
conversation. This becomes clear from the definitions of current density. Classi-
cally, current density is defined as in equation (2.17) and quantum mechanically
as in equation (2.18).
j =
I
A
= nv (2.17)
j = |A|2
v = v (2.18)
Equation (2.17) is the product of number density of particles passing through
unit area A with velocity v and equation (2.18) is the product of the probability
of the particle passing with the velocity v. Equation (2.16) is rewritten by
multiplying it with the velocity of the incident electron. The incident electron
is a plane wave as it is a free particle.
v =
kL
2m
(2.19)
kR
2m
|T|2
+
kL
2m
|R|2
=
kL
2m
(2.20)
Using equation (2.18) one obtains equation (2.21) which shows clearly that
current is conserved.
jT + jR = j0 (2.21)
Incident current is reflected back with probability |R|2
and transmitted
through the potential barrier with probability |T|2
, but no current is lost. Cur-
rent could be lost theoretically if it vanished inside the potential barrier. Equa-
tions (2.21) or (2.16) show that this is not possible.
The transmission coefficient can be entirely expressed by the electron’s momen-
tum k and its frequency ω
T =
4ikLkBe−ikRw
e−kBw (kB + ikR) (kB + ikL) − ekBw (kB − ikR) (kB − ikL)
(2.22)
T =
4ikB
kR
e−ikRw
ekBw 1 + ikB
kR
1 + ikB
kL
− e−kBw 1 − kB
kR
1 − kB
kL
(2.23)
7
8. |T|2
=
8 kB
kR
2
cosh (2kBw) 1 + kB
kR
2
1 + kB
kL
2
− 1 − kB
kR
2
1 − kB
kL
2
+
4k2
B
kLkR
(2.24)
Equation (2.24) can be written more compactly if incident and reflected
particle velocities are equal.
α =
kB
kL
=
kB
kR
(2.25)
|T|2
=
8α2
cosh (2kBw) (1 + α2)
2
− (1 − α2)
2
+ 4α2
(2.26)
The transmission probability through an asymmetric potential barrier decays
exponentially as the barrier width increases.
8
9. Chapter 3
Landauer Theory
In the free electron model the electrons are assumed to behave as free parti-
cles moving through a uniform potential formed by positively charged ions in
the lattice. Valence electrons in a metal are only weakly bound and interact
negligibly with each other and the ions in the lattice. The reason for the weak
interaction is that free electrons are plane waves which propagate freely in the
periodic structure of the ions in the lattice. The weak electron-electron inter-
action results from the Pauli exclusion principle which suppresses collisions of
electrons with each other and obey Fermi Dirac statistics. At equilibrium states
are filled according to the Fermi function
f0 (E − µ) =
1
1 + exp ((E − µ) /kBT)
(3.1)
At low temperatures the Fermi function approximates a step function
f0 (E) ≈ exp − (E − µ) /kBT (3.2)
As implied by equation (3.2) electrons will occupy states up to the Fermi
energy or chemical potential at very low temperatures. If instead of a constant
potential a periodic potential is assumed, the eigenstates of the electron have
the form of a plane wave times a function with the periodicity of the potential.
ϕnk = eikx
unk (x) (3.3)
In Landauer theory the single channel equation governing electron motion
in a conductor has a more general Hamiltonian, which also considers the effect
of magnetic fields in a conductor.
EC +
(i + eA)
2m
+ U (y) Ψ (x, y) = EΨ (x, y) (3.4)
For a transverse confining potential U (y) and magnetic field A = −ˆxBy ⇒
Ax = −ByandAy = 0
EC +
(i + eBy)
2m
+ U (y) Ψ (x, y) = EΨ (x, y) (3.5)
Ψ (x, y) =
1
√
L
exp (ikx) χ (y) (3.6)
9
10. For a confined potential in zero magnetic field the eigenfunctions are
Es +
2
k2
2m
+
p2
y
2m
+
1
2
mω2
0y2
χ (y) = Eχ (y) (3.7)
The eigenfunctions and eigenenergies are well-known
χn,k (y) = un (q) where q = mω0/ y (3.8)
E (n, k) = Es +
2
k2
2m
+ n +
1
2
ω0, n = 0, 1, 2, . . . (3.9)
un (q) = exp −q2
/2 Hn (q) (3.10)
Hn (q) are the nth Hermite polynomial. Dat97 The first three of these poly-
nomials are
H0 (q) =
1
π1/4
H1 (q) =
√
2q
π1/4
and H2 (q) =
2q2
− 1
√
2π1/4
(3.11)
In the Landauer approach current is based o the transmission of electrons
through the conductor. The current depends on the probability that an elec-
tron transmits from one contact to the other with respective chemical potentials
µ1 and µ2. Conductance, G (inverse of the resistance), of a large macroscopic
Figure 3.1: Transmission channel for quantum transport
conductor is directly proportional to its crosssectional area (A) and inversely
proportional to its length (L):
G = σA/L (3.12)
Dat04 Conductance would increase without boundary for smaller scales L. In-
stead it is found that a conductor of mesoscopic dimensions exhibits a maximum
conductance. This is puzzling as electrons suffer fewer collisions as distances are
scaled down. A conductor of mesoscopic size exhibits ballistic transport, but
still a finite conductance. At mesoscopic scales there are three characteristic
length scales:
10
11. 1. de Broglie wavelength
2. mean free path
3. phase relaxation length
An indication of what happens at these scales is the notion of two leads separated
by a channel. In the ballistic conductor few states exist, in fact if it is uncoupled
from the environment there is a single state governed by the single channel
equation. The contacts contain an infinite number of states posing no problem
for electrons to enter the contacts from the channel under the assumption that
they are reflectionless. Electrons are relocated in the system at transfer rates
which are determined by the average number of electrons in the channel and the
contacts. Electron transfer is induced as the contacts try to establish equilibrium
with the channel. The average number of electrons in any energy level in the
contacts is given by the Fermi function. Outflow and inflow of electrons from
one contact are given as γ1N and γ1f1. If Pauli blocking is considered the
transfer rates could also have been chosen as γ1f1 (1 − N) for the inflow and
γ1N (1 − f1) for the outflow. As the contacts try to establish equilibrium with
the channel the entire system departs from equilibrium resulting in a current
flow due to the difference in chemical potentials of the contacts.
µ1 − µ2 = qV (3.13)
The current from contact 1 towards the channel is sensibly given by the
difference in transfer rates.
I1 = (−q)
γ1
(f1 − N) (3.14)
I2 = (−q)
γ2
(f2 − N) (3.15)
In steady state there will be no net flux out or into the device:
I1 + I2 = 0 (3.16)
The average number of electrons in the channel in steady state is given by
N =
γ1f1 + γ2f2
γ1 + γ2
(3.17)
Substituting this expression in equations (3.14) and (3.15) gives the expres-
sion for the steady state current:
I = I1 = −I2 =
q γ1γ2
γ1 + γ2
[f1 ( ) − f2 ( )] (3.18)
An immediate consequence of equation (3.18) is that the energy levels lying
highly above or below the chemical potentials will not contribute to the current
as f1 ( ) = f2 ( ) = 0 is valid for high energy levels and f1 ( ) = f2 ( ) =
1 is valid for low energy levels. Equation (3.18) simplifies in the case of a
degenerate conductor with an energy level lying between the chemical potentials
of the contacts for which only states in a range of a few kBT at the Fermi level
contribute to the current:
11
12. I =
q γ1γ2
γ1 + γ2
(3.19)
For equal rate constants γ = γ1 = γ2 equation (3.19) simplifies to
I =
qγ
2
(3.20)
This expression is called the quantum of conductance. Equation (3.18) would
imply unlimited current as it does not take into consideration the effect of
broadening. Broadening describes the effect where a single state loses its defined
energy and decoheres into a set of states. In the channel attached to the source
and drain contacts the single state decoheres as the channel couples to the
contacts. As the state in the channel is influenced by its surroundings it acquires
a finite lifetime. The finite lifetime implies an uncertainty in the energy in
accordance with the uncertainty principle.
∆E ∝
1
τ
∝ γ (3.21)
The decoherence into a set of states centered around the chemical equilibrium
is a consequence of the uncertainty principle and and follows from a disturbance
of the pure state from the surrounding noise. Then the coupling constant is a
measure of the noise as it determines by the reciprocal value the finite lifetime
of the coupled states in the channel.
The spread of states around the chemical potential surpasses the energy window
defined by the range lying between the chemical potentials µ1 and µ2 of source
and drain contacts. It is proportional to to the coupling constant. States lying
outside of this energy window are not current carrying states. Current through
the channel is therefore reduced by the fraction of the energy window width and
coupling constant times a proportionality factor. The uncertainty in energy
introduced by the disturbance from noise reduces the current. On the other
hand the creation of additional states due to the uncertainty principle makes it
possible for electrons from the contacts to occupy empty states in the energy
window in the channel. If there was only a single state Pauli blocking would
prevent electrons from transferring from the contacts into the channel’s states.
Therefore the uncertainty relation is necessary for current flow but is responsible
for the reduction of the current as it is dependent on the coupling strength given
by the rate coefficient γ.
(µ1 − µ2)
Cγ1
< 1 (3.22)
The fraction of conducting channels over the total number of energy levels
according to the energy spread δE must be of course less than one. Cγ1 is the
effective width of the broadened channel where C is a numerical constant.
∆Eδt =
γ
γ = (3.23)
Indeed, the product of the lifetime of a state and its spread in energy is equal
to .
12
13. The broadened density of states can be described by a Lorentzian centered
around E =
D =
γ/2π
(E − )
2
+ (γ/2)
2 (3.24)
The Landauer-Buettiker formula includes the new density of states distribu-
tion due to broadening to calculate the current:
I =
q
dED (E)
γ1γ2
γ1 + γ2
[f1 ( ) − f2 ( )] (3.25)
One can write this in the form:
I =
q ∞
−∞
¯T (E) [f1 (E) − f2 (E)] (3.26)
¯T (E) ≡ 2πD (E)
γ1γ2
γ1 + γ2
=
γ1γ2
(E − )
2
+ (γ/2)
2 (3.27)
For a small bias the density of states in the contacts can be assumed to be
constant:
I =
q
[µ1 − µ2]
γ1γ2
(µ − )
2
+ ((γ1 + γ2) /2)
2 (3.28)
The maximum current is obtained if the energy level coincides with µ, the
average of µ1 and µ2. From equation (3.13) the maximum conductance is
G ≡
I
VD
=
q2
4γ1γ2
(γ1 + γ2)
2 =
q2
if γ1 = γ2 (3.29)
The current through the conductor depends on the probability that an elec-
tron is transmitted. The transmission probability is a resonance curve as can
be seen from the quadratic terms in the denominator in equation (3.27). Res-
onance occurs if the energy level is equal to the chemical potential. However,
broadening implies that there is maximum conductance. The effect of broaden-
ing is described by the Lorentzian density of state. The maximum conductance
and transmission resonance are not contradictory.
13
14. Chapter 4
Tight binding wires
The motivation for the treatment of tight binding wires comes from the pos-
sibility to derive the transmission resonance similar to the Landauer formula
from the tight binding method. In tight binding theory the atomic orbitals are
eigenfunctions of the Hamiltonian of a single atom and have a cutoff at the
lattice constant. If atoms are arranged on a linear chain and are brought suf-
ficiently near to each other, electron transfer from one atom to its neighbour
becomes possible by tunneling. This happens by transferring from a filled state
to an empty state of an adjacent atom when the density of states overlap. A
tight binding wire is a one dimensional chain of atoms with onsite energies and
hopping integrals.
Figure 4.1: Tight binding wire with onsite energies a and hopping integrals b
Hαβ =
aα, β = α
bβ, β = α + 1
bβ, β = α − 1
0, otherwise
(4.1)
For a chain with 4 atoms the Hamiltonian matrix is:
Hαβ
a b 0 0 0
b a b 0 0
0 b a b 0
0 0 b a b
(4.2)
For interatomic distances it will be possible for electrons to move from one
atom to another. This can be pictured as the transfer of the electron from
one atomic orbital ϕα to adjacent ones ϕα+1 and ϕα−1. The eigenstates of the
Hamiltonian can therefore be written as a linear superposition of the individual
atomic states.
|ϕ = Cα|α (4.3)
14
15. The Schroedinger equation becomes
β
HαβCβ = ECα (4.4)
This expression can be expanded according to the rules given in equation
(4.1).
β
HαβCβ =
β
HαβCβ +
β
Hαβ+1Cβ+1 +
β
Hαβ−1Cβ−1 (4.5)
β
HαβCβ = aCα + bCα+1 + bCα−1 (4.6)
The nearest neighbour model is compactly expressed in equation (4.7).
0 = (E − aα) Cα − bα+1Cα+1 − bα−1Cα−1 (4.7)
From
i
∂
∂t
ϕ = Hϕ (4.8)
the probability amplitudes are related by the set of N coupled equations.
i
dCα
dt
= aCα + bCα+1 + bCα−1 (4.9)
If the hopping integrals were zero, equation (4.9) would reduce to equation
(4.10) indicating the oscillating time dependence of the probability amplitude
Cα.
i
dCα
dt
= aCα (4.10)
The wavefunction for the electron is then given in equation (4.11) confirming
that a is the energy of the state ϕα, or atomic onsite energy.
Cα = exp (−iat/ ) (4.11)
As a result of the translational invariance, Bloch’s theorem becomes applica-
ble, and the solutions for Cα are running wave solutions. In general the solution
will be a superposition of forward and backward traveling waves.
Cα = A exp i kα −
at
+ B exp −i kα +
at
(4.12)
The time dependence can be neglected of the three cases considered, as it
can be canceled out in the calculation. Thus Cα takes the form
Cα = A exp (kα) + B exp (−ikα) (4.13)
In the tight binding wire the potential barrier problem can be investigated
in a more general context. Electrons travel in the form of plane waves up and
down the linear chain of atoms and experience potential barriers depending on
the conditions specified by the hopping integrals and onsite energies on the wire.
Three cases are considered:
15
16. 1. The perfect infinite wire
2. The perfect semi-infinite wire
3. The infinite wire with a defect
4.1 The perfect infinite wire
In the case of a perfect infinite wire all onsite energies and hopping integrals
become equal.
aα = a ∀α (4.14)
bα = b ∀α (4.15)
Inserting equation (4.13) into equation (4.9), with equal onsite energies and
hopping integrals everywhere in the chain gives
E A exp (ikα)+B exp (−ikα) = aCα+b A exp (ikα)+B exp (−ikα) exp (ikα)+exp (−ikα)
(4.16)
This gives the dispersion relation
ECα = (a + 2b cos (kα)) Cα (4.17)
4.2 The perfect semi-infinite wire
For the perfect semi-infinite wire all onsite energies and hopping integrals are
equal except for bL = bR = 0, introducing a break in the wire. The matrix
representation of the tight binding Hamiltonian for four atoms is:
Hαβ =
a 0 0 0
0 a 0 0
0 b a 0
0 0 b a
(4.18)
Expanding equation (4.4) with the matrix representation for four atoms
gives:
β
HαβCβ =
a 0 0 0
0 a 0 0
0 b a 0
0 0 b a
C0
C1
C2
C3
=
aC0
aC1 + bC2
aC2 + bC3 + bC1
aC3 + bC2 + bC4
(4.19)
EC0 = aC0 (4.20)
EC1 = aC1 + bC2 (4.21)
EC2 = aC2 + bC3 + bC1 (4.22)
EC3 = aC3 + bC2 (4.23)
16
17. For α = 0 the eigenvalue E is the onsite energy. This makes sense for
the isolated atom introduced as a break in the wire. In the case of α = 1 an
eigenvalue is not easily found. By substituting equation (4.13) into equation
(4.21) one gets:
0 = (E − a) A exp (ikα) + B exp (−ikα) − b A exp (2ikα) + B exp (−2ikα)
(4.24)
0 = b A exp (2ikα)+B +A+B exp (−2ikα) −b A exp (2ik)+B exp (−2ikα)
(4.25)
0 = b (A + B) (4.26)
B = −A (4.27)
This is equivalent to an infinite potential separating the wire into two halves.
The wave is then reflected at the barrier. As a result Cα is a superposition of
a forward and a reflected plane wave. For α > 1 the eigenvalue equations have
the same pattern which can be generalized to the expression given in equation
(4.17). We get from
ECα = aCα + 2b cos (kα) Cα
E − a = 2b cos (kα) (4.28)
4.3 The infinite wire with a defect
The Hamiltonian for the infinite wire corresponds to figure 4.1 and has condi-
tions:
aα = aL ∀α ≤ −1 (4.29)
a0 = 0 (4.30)
aα = aR ∀α ≥ 1 (4.31)
bα = b ∀α < −1 ∧ α ≥ 1 (4.32)
b−1 = bL (4.33)
b0 = bR (4.34)
The eigenstates of this Hamiltonian are of the type
Cα = exp (ikLα) + R exp (−ikLα) ∀α ≤ −1 (4.35)
Cα = T exp (ikRα) ∀α ≥ 1 (4.36)
It is verified separately for ∀α < −1 and ∀α > 1 that equations (4.35) and
(4.36) satisfy the nearest neighbour model.
∀α < −1
17
18. (E − aL) Cα − bCα+1 − bCα−1 = 0 (4.37)
(E − aL) eikLα
+Re−ikLα
−b eikL(α+1)
+Re−ikL(α+1)
−b eikL(α−1)
+Re−ikL(α−1)
= 0
(4.38)
(E − aL) eikLα
+Re−ikLα
−beikLα
eikLα
+e−ikLα
−bRe−ikLα
eikLα
+e−ikLα
= 0
(4.39)
(E − aL) eikLα
+ Re−ikLα
+ 2b cos (kLα) eikLα
+ Re−ikLα
= 0 (4.40)
E = aL + 2b cos (kL) (4.41)
∀α > 1
(E − aR) Cα − bCα+1 − bCα−1 = 0 (4.42)
(E − aR) TeikRα
− b TeikR(α+1)
− b TeikR(α−1)
= 0 (4.43)
E = aR + 2b cos (kRα) (4.44)
The unknown reflection and transmission coefficients R and T are deter-
mined by setting α = −1 to find R, and α = 1 to find T in the nearest neighbour
equation (4.7),
For α = −1
b e−2ikLα
+ Re2ikL
+ R + 1 − bLC0 − b e−2ikLα
+ Re−2ikLα
= 0 (4.45)
b e−2ikLα
+ Re2ikLα
− b e−2ikLα
+ Re2ikLα
+ b (R + 1) − bLC0 = 0 (4.46)
bLC0 = b (R + 1) (4.47)
R =
bL
b
C0 − 1 (4.48)
For α = 1
b eikRα
+ e−ikRα
TeikRα
− bTe2ikRα
− bRC0 = 0 (4.49)
18
19. bTe2ikRα
+ bT − bTe2ikRα
− bRC0 = 0 (4.50)
T =
bR
b
C0 (4.51)
Setting α = 0 and inserting the expressions for R and T an expression for
C0 is found.
For α = 0
(E − a) C0 − bC1 − bLC−1 = 0 (4.52)
(E − a) C0 − bTeikR
− bL e−ikL
+ ReikL
= 0 (4.53)
(E − a) C0 − bReikR
C0 −
b2
L
b
eikL
C0 + bLeikL
− bLe−ikL
= 0 (4.54)
E − a − bReikR
−
b2
L
b
eikL
C0 + bL eikL
− e−ikL
= 0 (4.55)
C0 =
−2bLi sin (kL)
E − a − bReikR −
b2
L
b eikL
(4.56)
The transmission coefficient T squared is the transmission probability.
|T|2
=
bR
b
2
C0C∗
0 (4.57)
|T|2
=
2bLbR
b
2
sin2
(kL)
(E − a)
2
− 2 (E − a) bR cos (kR) +
b2
L
b cos (kL) +
b2
L
bR
b cos (kR − kL) + b2
R +
b4
L
b2
(4.58)
Equation (4.58) displays a resonance curve similarly to the transmission
function in the Landauer approach. Transmission peaks at a defined energy
which is the atomic or molecular onsite energy of the defect.
19
20. Chapter 5
The Scanning Tunneling
Microscope (STM)
The operation of a scanning tunneling microscope relies on the quantum me-
chanical effect of tunneling from an atomically sharp probe to the surface en-
abling atomic resolutions of the surface. The tip is brought near enough to
the surface in order to obtain a measurable tunneling current between tip and
surface. Electrons tunnel from the tip to the surface depending on the polar-
ity of the applied voltage between surface and tip. The tip scans the surface
in two dimensions which produces a contour map of the surface as the height
is adjusted to maintain a constant tunneling current. The tunneling current
decays exponentially with the vacuum gap distance between tip and surface.
This result was shown for the asymmetric barrier case. The tunneling current
between surface and tip is calculated by Bardeen’s formalism
I =
2π
f (Eµ) [1 − f (Eν + eV )] |Mµν|2
δ (Eµ − Eν) (5.1)
This result looks familiar from Landauer theory where |Mµν|2
is the tun-
neling matrix element between states ϕµ and ϕν. f (Eµ) [1 − f (Eν + eV )] is
the probability that an electron will tunnel from a filled into an empty state.
δ (Eµ − Eν) expresses the width of the energy window between states which
contribute to the current. At small voltages and low temperature the Fermi
function becomes a delta function
I =
2π
e2
V
µν
|Mµν|2
δ (Enu − EF ) δ (Emu − EF ) (5.2)
This equation implies that the tunneling current is proportional to the local
density of states at the Fermi energy.
Transmission resonance can occur when electrons tunnel through a molecule
which is represented as a defect in the tight binding wire approach. Similarly
when a molecule is placed between the tip of a STM and the surface trans-
mission resonance becomes possible. This situation is found often as molecules
adsorbed on the surface between the tip and the surface. An electron tunneling
20
21. from the tip to the surface can be modeled as an electron hopping to a neigh-
bouring site. Tunneling from atom aL to aR corresponds to tunneling through
an asymmetric potential barrier where VL = aL and VR = aR.
21
22. Chapter 6
Tight binding wire with an
oscillator
The atom at site 0 will now be allowed to oscillate. The Hamiltonian now
becomes
ˆH = ˆHe + ˆHo + ˆHe−o (6.1)
where ˆHe is the tight binding Hamiltonian for the electrons, ˆHo = ˆa†
ˆa + 1
2 ω
where ω is the oscillator frequency, and ˆa†
and ˆa are the raising and lowering
operators respectively, and ˆHe−o = λ ˆa†
+ ˆa |0 0| where λ is the strength of
the coupling between the electron and the oscillator. Let the eigenstates of ˆHo
be |n so that
ˆHo|n = Wn|n = n +
1
2
ω|n (6.2)
The wavefunction for the system is now expanded in the atomic orbitals |α
and the oscillator states |n giving
|ψ =
βm
Cβm|βm (6.3)
From Schroedinger’s equation we then get
ˆH|ψ = E|ψ
⇒
βm
αn| ˆHe + ˆHo + ˆHe−o|βm Cβm = ECαn
⇒
βm
Hαβδmn + Wnδmnδαβ + λδα0δβ0
√
nδn,m+1 +
√
n + 1δn,m−1 Cβm = ECαn
⇒
β
HαβCβn + λδα0
√
nC0n−1 +
√
n + 1C0n+1 = (E − Wn) Cαn
Making use of the fact that our tight binding Hamiltonian only involves
nearest neighbours one gets
22
23. (E − Wn − aα) Cαn−bαCα+1n−bα−1Cα−1n−λδα0
√
nC0n−1 +
√
n + 1C0n+1 = 0
(6.4)
It can be shown that equation (6.4) is satisfied by equations (6.5) and (6.6)
Cαn = In exp (ikL,nα) + Rn exp (−ikL,nα) for α ≤ −1 (6.5)
Cαn = Tn exp (ikR,nα) for α ≥ 1 (6.6)
Considering α = −1 in equation (6.4) it can be shown that
Rn = (bL/b) C0n − In (6.7)
Considering α = 1 in equation (6.4) it can be shown that
Tn = (bR/b) C0n (6.8)
Considering α = 0 it can be shown that n AmnC0n = Bm where
Bm = −2iIm sin (kL,m) (6.9)
Amn =
E − Wn − a −
b2
R
b exp (ikR,n) −
b2
L
b exp (ikR,n) , n = m
−λ
√
n, n = m + 1
−λ
√
n + 1, n = m − 1
0, otherwise
(6.10)
Consider the special case where we only allow the oscillator to have zero or
one phonons and where I0 = 1 and I1 = 0. In this case one can find C00 and
C01 analytically, and from this get transmissions T0 and T1.
23
24. Bibliography
[Dat97] Datta,S., "Electronic Transport in Mesoscopic Systems" Cambridge
University Press (1997)
[Dat04] Datta,S.,"Electrical Resistance: an atomistic view" Nanotechnology 15
(2004), 433-451
24
