Electron transport in one-dimensional nanoscalesystems

Benjamin Yiwen Faerber

May 25, 2014

Contents

1 Introduction 3

2 Asymmetric potential barrier 5

3 Landauer Theory 9

4 Tight binding wires 144.1 The perfect infinite wire . . . . . . . . . . . . . . . . . . . . . . . 164.2 The perfect semi-infinite wire . . . . . . . . . . . . . . . . . . . . 164.3 The infinite wire with a defect . . . . . . . . . . . . . . . . . . . . 17

5 The Scanning Tunneling Microscope (STM) 20

6 Tight binding wire with an oscillator 22

2

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 themesoscopic scale. Devices like the scanning tunneling microscope are able toattain atomic resolutions by the quantum mechanical effect of tunneling. Thisrepresents a great motivation to explore the connection between the mechanismof conduction on mesoscopic scales and its implications for tunneling through avacuum 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 anelectron will be transmitted through a conductor. The approach is quantummechanical and not semi classical as individual electrons are confined to movein single channels. Transport becomes ballistic instead of diffusive as scatteringfrom phonons and impurities becomes negligible on mesoscopic scales. An elec-tric field acting as a force carrier becomes less relevant. Instead the emphasis isshifted 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 asymmetricpotential barrier is thus investigated as it provides a simple model of transportthrough the channel. As a next step a linear chain of atoms representing aone dimensional conductor is treated as a tight binding wire. In tight bindingtheory the crystal Hamiltonian consists of the atomic Hamiltonian and a correc-tion term which produces a periodic potential with the periodicity of the Bravaislattice. Eigenstates of the tight binding Hamiltonian are superpositions of theatomic orbitals. The constitution of the wire is then varied for three cases. Inthe simplest case all atoms have equal onsite energies as well as equal hoppingintegrals 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 adifferent onsite energy than the rest of the atoms in the chain. The nanowireis studied as it provides a good understanding of the case of electron trans-mission through an atom or molecule. Overlapping atomic orbitals imply thatelectrons have a non zero velocity throughout the atomic chain. The hoppingintegrals imply therefore electron transport via tunneling between neighbouringatomic sites where the onsite energies of the atoms represent the potential bar-

3

rier height. Effectively this is equivalent to the asymmetric potential model. Intheory the transmission of particles through a potential barrier serves as a goodunderstanding of the one dimensional case of tunneling through a vacuum gap.The remarkable application of this model is the scanning tunneling microscopewhere tunneling through a vacuum gap is its central mechanism. In the moreextensive 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 modelleaves open the possibility of an adsorbed molecule on the surface. The studyof adsorbate molecules under STM operation reveals their electronic stuctureas transmission resonance is observed. In the case of adsorbate vibrations con-duction peaks are observed corresponding to correlations between vibrationalmodes and transmission. Possibly these are the results of contributing inelasticchannels in the conduction process. The interpretation is referred from Mingoet al. who consider the probabiliy of excitation of vibrational modes under STMoperation as related to the lifetime of the vibration due to electron hole pairexcitations.

4

Chapter 2

Asymmetric potentialbarrier

Electrons incident on the left are free particles. The incident electron is thus aplane wave travelling in the positive x direction. At x = 0 there is the probabilitythat the particle is reflected back which results in a plane wave of amplitudeR (reflection coefficient) travelling backwards. The wave function in the regionx < 0 is thus a superposition of two plane waves travelling in the positive andnegative direction. In the potential barrier region 0 < x < w the particle is in aconfined space resulting in a standing wave. On the right side of the potentialbarrier x > w the electron is a free particle represented by a plane wave withamplitude T travelling in the positive x direction. The squared reflection andtransmission coefficient give the probabilities of the particle being reflected ortransmitted 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 derivativesmust be piecewise continuous over region one, two and three. Applying theseat 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 matrixequation (2.6) is obtained with the vector containing coefficients R and T whichis inverted, yielding direct expressions for the coefficients R and T only in termsof the wave vectors.

ikL (1−R) = kB (A+B) (2.4)

AekBw +Be−kBw = TeikBw (2.5)

5

Figure 2.1: Asymmetric potential barrier

(kB + ikLkB − ikL

)=(− (kB − ikL) e−kBweikRw (kB + ikR)− (kB + ikL) ekBweikRw (kB − ikR)

)(RT

)(2.6)

(RT

)= 1

∆

(e−kBweikL (kB − ikR) −e−kBweikRw (kB + ikR)

(kB + ikL) − (kB − ikL)

)(kB + ikLkB − ikL

)(2.7)

∆ =∣∣∣∣− (kB − ikL) e−kBweikRw (kB + ikR)− (kB + ikL) ekBweikRw (kB − ikR)

∣∣∣∣ (2.8)

∆ = 2eikRw((kRkL − k2

B

)sinh (kBw) + ikB (kL + kR) cosh (kBw)

)(2.9)

R = 1∆[ekBweikRw (kB − ikR) (kB + ikL)− e−kBweikRw (kB + ikR) (kB − ikL)

](2.10)

R = 2eikRw

∆[(k2B + 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

[(k2B + kLkR

)2 sinh2 (kBw) + (kLkB − kRkB)2 cosh2 (kBw)]

(2.13)

6

|T |2 = 16|∆|2 (kBkL)2 (2.14)

|R|2 + kRkL|T |2 = 4

|∆|2

[4k2BkRkL +

(k2B − kRkL

)2 sinh2 (kBw) + (kBkL + kBkR)2 cosh2 (kBw)

+4k2BkLkR sinh2 (kBw)− 4k2

BkRkL cosh2 (kBw)]

(2.15)

kRkL|T |2 + |R|2 = 1 (2.16)

Equation (2.16) is the central result of this calculation and expresses currentconversation. This becomes clear from the definitions of current density. Classi-cally, current density is defined as in equation (2.17) and quantum mechanicallyas in equation (2.18).

j = I

A= nv (2.17)

j = |A|2v = v (2.18)

Equation (2.17) is the product of number density of particles passing throughunit area A with velocity v and equation (2.18) is the product of the probabilityof the particle passing with the velocity v. Equation (2.16) is rewritten bymultiplying it with the velocity of the incident electron. The incident electronis a plane wave as it is a free particle.

v = ~kL2m (2.19)

~kR2m |T |

2 + ~kL2m |R|

2 = ~kL2m (2.20)

Using equation (2.18) one obtains equation (2.21) which shows clearly thatcurrent is conserved.

jT + jR = j0 (2.21)

Incident current is reflected back with probability |R|2 and transmittedthrough 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

kRe−ikRw

ekBw(

1 + ikB

kR

)(1 + ikB

kL

)− e−kBw

(1− kB

kR

)(1− kB

kL

) (2.23)

7

|T |2 =8(kB

kR

)2

cosh (2kBw)(

1 +(kB

kR

)2)(

1 +(kB

kL

)2)−(

1−(kB

kR

)2)(

1−(kB

kL

)2)

+ 4k2B

kLkR

(2.24)Equation (2.24) can be written more compactly if incident and reflected

particle velocities are equal.

α = kBkL

= kBkR

(2.25)

|T |2 = 8α2

cosh (2kBw) (1 + α2)2 − (1− α2)2 + 4α2(2.26)

The transmission probability through an asymmetric potential barrier decaysexponentially as the barrier width increases.

8

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 inthe lattice. Valence electrons in a metal are only weakly bound and interactnegligibly with each other and the ions in the lattice. The reason for the weakinteraction is that free electrons are plane waves which propagate freely in theperiodic structure of the ions in the lattice. The weak electron-electron inter-action results from the Pauli exclusion principle which suppresses collisions ofelectrons with each other and obey Fermi Dirac statistics. At equilibrium statesare filled according to the Fermi function

f0 (E − µ) = 11 + 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 Fermienergy or chemical potential at very low temperatures. If instead of a constantpotential a periodic potential is assumed, the eigenstates of the electron havethe form of a plane wave times a function with the periodicity of the potential.

ϕnk = eikxunk (x) (3.3)

In Landauer theory the single channel equation governing electron motionin a conductor has a more general Hamiltonian, which also considers the effectof 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 = −xBy ⇒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

For a confined potential in zero magnetic field the eigenfunctions are[Es + ~2k2

2m +p2y

2m + 12mω

20y

2]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 + ~2k2

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) = 1π1/4 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 electronsthrough the conductor. The current depends on the probability that an elec-tron transmits from one contact to the other with respective chemical potentialsµ1 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 inverselyproportional 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 maximumconductance. This is puzzling as electrons suffer fewer collisions as distances arescaled down. A conductor of mesoscopic size exhibits ballistic transport, butstill a finite conductance. At mesoscopic scales there are three characteristiclength scales:

10

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 separatedby a channel. In the ballistic conductor few states exist, in fact if it is uncoupledfrom the environment there is a single state governed by the single channelequation. The contacts contain an infinite number of states posing no problemfor electrons to enter the contacts from the channel under the assumption thatthey are reflectionless. Electrons are relocated in the system at transfer rateswhich are determined by the average number of electrons in the channel and thecontacts. Electron transfer is induced as the contacts try to establish equilibriumwith the channel. The average number of electrons in any energy level in thecontacts is given by the Fermi function. Outflow and inflow of electrons fromone contact are given as γ1N and γ1f1. If Pauli blocking is considered thetransfer rates could also have been chosen as γ1f1 (1−N) for the inflow andγ1N (1− f1) for the outflow. As the contacts try to establish equilibrium withthe channel the entire system departs from equilibrium resulting in a currentflow 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 thedifference 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 (ε)− f2 (ε)] (3.18)

An immediate consequence of equation (3.18) is that the energy levels lyinghighly above or below the chemical potentials will not contribute to the currentas f1 (ε) = f2 (ε) = 0 is valid for high energy levels and f1 (ε) = f2 (ε) =1 is valid for low energy levels. Equation (3.18) simplifies in the case of adegenerate conductor with an energy level lying between the chemical potentialsof the contacts for which only states in a range of a few kBT at the Fermi levelcontribute to the current:

11

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) wouldimply unlimited current as it does not take into consideration the effect ofbroadening. Broadening describes the effect where a single state loses its definedenergy and decoheres into a set of states. In the channel attached to the sourceand drain contacts the single state decoheres as the channel couples to thecontacts. As the state in the channel is influenced by its surroundings it acquiresa finite lifetime. The finite lifetime implies an uncertainty in the energy inaccordance with the uncertainty principle.

∆E ∝ 1τ∝ γ (3.21)

The decoherence into a set of states centered around the chemical equilibriumis a consequence of the uncertainty principle and and follows from a disturbanceof the pure state from the surrounding noise. Then the coupling constant is ameasure of the noise as it determines by the reciprocal value the finite lifetimeof the coupled states in the channel.The spread of states around the chemical potential surpasses the energy windowdefined by the range lying between the chemical potentials µ1 and µ2 of sourceand drain contacts. It is proportional to to the coupling constant. States lyingoutside of this energy window are not current carrying states. Current throughthe channel is therefore reduced by the fraction of the energy window width andcoupling constant times a proportionality factor. The uncertainty in energyintroduced by the disturbance from noise reduces the current. On the otherhand the creation of additional states due to the uncertainty principle makes itpossible for electrons from the contacts to occupy empty states in the energywindow in the channel. If there was only a single state Pauli blocking wouldprevent electrons from transferring from the contacts into the channel’s states.Therefore the uncertainty relation is necessary for current flow but is responsiblefor the reduction of the current as it is dependent on the coupling strength givenby the rate coefficient γ.

(µ1 − µ2)Cγ1

< 1 (3.22)

The fraction of conducting channels over the total number of energy levelsaccording to the energy spread δE must be of course less than one. Cγ1 is theeffective width of the broadened channel where C is a numerical constant.

∆Eδt = ~γγ = ~ (3.23)

Indeed, the product of the lifetime of a state and its spread in energy is equalto ~.

12

The broadened density of states can be described by a Lorentzian centeredaround E = ε

Dε = γ/2π(E − ε)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

~

∫dEDε (E) γ1γ2

γ1 + γ2[f1 (ε)− f2 (ε)] (3.25)

One can write this in the form:

I = q

~

∫ ∞−∞

T (E) [f1 (E)− f2 (E)] (3.26)

T (E) ≡ 2πDε (E) γ1γ2

γ1 + γ2= γ1γ2

(E − ε)2 + (γ/2)2 (3.27)

For a small bias the density of states in the contacts can be assumed to beconstant:

I = q

~[µ1 − µ2] γ1γ2

(µ− ε)2 + ((γ1 + γ2) /2)2 (3.28)

The maximum current is obtained if the energy level ε coincides with µ, theaverage 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 canbe 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 conductanceand transmission resonance are not contradictory.

13

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 formulafrom the tight binding method. In tight binding theory the atomic orbitals areeigenfunctions of the Hamiltonian of a single atom and have a cutoff at thelattice 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 neighbourbecomes possible by tunneling. This happens by transferring from a filled stateto an empty state of an adjacent atom when the density of states overlap. Atight binding wire is a one dimensional chain of atoms with onsite energies andhopping integrals.

Figure 4.1: Tight binding wire with onsite energies a and hopping integrals b

Hαβ =

aα, β = αbβ , β = α+ 1bβ , β = α− 10, otherwise

(4.1)

For a chain with 4 atoms the Hamiltonian matrix is:

Hαβ

a b 0 0 0b a b 0 00 b a b 00 0 b a b

(4.2)

For interatomic distances it will be possible for electrons to move from oneatom to another. This can be pictured as the transfer of the electron fromone atomic orbital ϕα to adjacent ones ϕα+1 and ϕα−1. The eigenstates of theHamiltonian can therefore be written as a linear superposition of the individualatomic states.

|ϕ〉 =∑

Cα|α〉 (4.3)

14

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~∂

∂tϕ = Hϕ (4.8)

the probability amplitudes are related by the set of N coupled equations.

i~dCαdt

= 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 amplitudeCα.

i~dCαdt

= aCα (4.10)

The wavefunction for the electron is then given in equation (4.11) confirmingthat 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 solutionwill 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 itcan 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 anddown the linear chain of atoms and experience potential barriers depending onthe conditions specified by the hopping integrals and onsite energies on the wire.Three cases are considered:

15

1. The perfect infinite wire

2. The perfect semi-infinite wire

3. The infinite wire with a defect

4.1 The perfect infinite wireIn the case of a perfect infinite wire all onsite energies and hopping integralsbecome equal.

aα = a ∀α (4.14)bα = b ∀α (4.15)

Inserting equation (4.13) into equation (4.9), with equal onsite energies andhopping 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 wireFor the perfect semi-infinite wire all onsite energies and hopping integrals areequal except for bL = bR = 0, introducing a break in the wire. The matrixrepresentation of the tight binding Hamiltonian for four atoms is:

Hαβ =

a 0 0 00 a 0 00 b a 00 0 b a

(4.18)

Expanding equation (4.4) with the matrix representation for four atomsgives:

∑β

HαβCβ =

a 0 0 00 a 0 00 b a 00 0 b a

C0C1C2C3

=

aC0

aC1 + bC2aC2 + bC3 + bC1aC3 + 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

For α = 0 the eigenvalue E is the onsite energy. This makes sense forthe isolated atom introduced as a break in the wire. In the case of α = 1 aneigenvalue 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 ofa forward and a reflected plane wave. For α > 1 the eigenvalue equations havethe 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 defectThe 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

(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 neighbourequation (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 = bLbC0 − 1 (4.48)

For α = 1

b(eikRα + e−ikRα

)TeikRα − bTe2ikRα − bRC0 = 0 (4.49)

18

bTe2ikRα + bT − bTe2ikRα − bRC0 = 0 (4.50)

T = bRbC0 (4.51)

Setting α = 0 and inserting the expressions for R and T an expression forC0 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 − bReikRC0 −b2LbeikLC0 + bLe

ikL − bLe−ikL = 0 (4.54)

(E − a− bReikR − b2L

beikL

)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 =(bRb

)2C0C

∗0 (4.57)

|T |2 =(

2bLbRb

)2 sin2 (kL)(E − a)2 − 2 (E − a)

(bR cos (kR) + b2

L

b cos (kL))

+ b2LbR

b cos[

(kR − kL)]

+ b2R + b4L

b2

(4.58)Equation (4.58) displays a resonance curve similarly to the transmission

function in the Landauer approach. Transmission peaks at a defined energywhich is the atomic or molecular onsite energy of the defect.

19

Chapter 5

The Scanning TunnelingMicroscope (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 tothe surface in order to obtain a measurable tunneling current between tip andsurface. 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 surfacein two dimensions which produces a contour map of the surface as the heightis adjusted to maintain a constant tunneling current. The tunneling currentdecays exponentially with the vacuum gap distance between tip and surface.This result was shown for the asymmetric barrier case. The tunneling currentbetween surface and tip is calculated by Bardeen’s formalism

I = 2π~∑

f (Eµ) [1− f (Eν + eV )] |Mµν |2δ (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 )] isthe 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 whichcontribute to the current. At small voltages and low temperature the Fermifunction becomes a delta function

I = 2π~e2V

∑µν

|Mµν |2δ (Enu − EF ) δ (Emu − EF ) (5.2)

This equation implies that the tunneling current is proportional to the localdensity of states at the Fermi energy.

Transmission resonance can occur when electrons tunnel through a moleculewhich is represented as a defect in the tight binding wire approach. Similarlywhen a molecule is placed between the tip of a STM and the surface trans-mission resonance becomes possible. This situation is found often as moleculesadsorbed on the surface between the tip and the surface. An electron tunneling

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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 throughan asymmetric potential barrier where VL = aL and VR = aR.

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Chapter 6

Tight binding wire with anoscillator

The atom at site 0 will now be allowed to oscillate. The Hamiltonian nowbecomes

H = He + Ho + He−o (6.1)

where He is the tight binding Hamiltonian for the electrons, Ho =(a†a+ 1

2)~ω

where ω is the oscillator frequency, and a† and a are the raising and loweringoperators respectively, and He−o = λ

(a† + a

)|0〉〈0| where λ is the strength of

the coupling between the electron and the oscillator. Let the eigenstates of Ho

be |n〉 so that

Ho|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

H|ψ〉 = E|ψ〉⇒∑βm

〈αn|He + Ho + He−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 involvesnearest neighbours one gets

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(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∑nAmnC0n = Bm where

Bm = −2iIm sin (kL,m) (6.9)

Amn =

E −Wn − a− b2

R

b exp (ikR,n)− b2L

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 orone phonons and where I0 = 1 and I1 = 0. In this case one can find C00 andC01 analytically, and from this get transmissions T0 and T1.

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Bibliography

[Dat97] Datta,S., "Electronic Transport in Mesoscopic Systems" CambridgeUniversity Press (1997)

[Dat04] Datta,S.,"Electrical Resistance: an atomistic view" Nanotechnology 15(2004), 433-451

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