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Electrons in Nanostructures Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. • Electrons travel at a drift velocity depending on the applied voltage (Ohm’s law). • The scattering events that contribute to resistance occur with mean free paths that are typically tens of nm in many metals. (1 every 40 nm in Cu). If the size of the structure is of the same scale as the mean free path of an electron, Ohm’s law may not

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Page 1: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

Electrons in NanostructuresElectrons in Nanostructures

• The electronic properties of bulk materials are dominated by electron scattering.

• Electrons travel at a drift velocity depending on the applied voltage (Ohm’s law).

• The scattering events that contribute to resistance occur with mean free paths that are typically tens of nm in many metals. (1 every 40 nm in Cu).

If the size of the structure is of the same scale as the mean free path of an electron, Ohm’s law may not apply, giving rise to quantum effects [quantum confinement in 1D (nanowires) and 0D (quantum dots)].

Page 2: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

Electronic properties of materialsElectronic properties of materials

A

LR

Copyright Stuart Lindsay 2008

L

A

24 orders of magnitude between copper and rubber!

resistivity

Page 3: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

• Individual electrons interact strongly

• Excited carriers behave like “free electrons” owing to screening by “correlation hole”

The Fermi Liquid

Fermi energyFermi energy = energy of the highest occupied state at zero temperature.= chemical potential at T=0

Page 4: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

*

22

2m

kE

The energy of the quasi-particle can be written as:

m*m* = effective mass

A typical Fermi energy is 200 times the thermal energy.

Mobile electrons are produced from thermal fluctuations that promote electrons from below the Fermi energy to above it.

Carriers are not produced alone, but in correlated pairs (electron/holeelectron/hole), so that Coulomb interactions may be ignored.

These correlated pairs can be treated as quasi-particles in These correlated pairs can be treated as quasi-particles in stationary states of the system!stationary states of the system!

Page 5: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

Drude free-electron modelDrude free-electron model

Ev

em

vne EJ

mean time between collisions

Ohms Law:

v drift velocity

From Newton’s law:

Conduction electrons are a gas of free, non-interacting particles, exchanging energy only by scattering events.

conductivity

Page 6: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

m

ne

2

1

ρ(Cu)= 1.6 micro Ohm·cm =2.7·10-14 s (correct)

But:

nmv

l 1

m

kTv

3

The electron mean free path in copper is ≈ 40nm.

Page 7: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

Sommerfeld ModelSommerfeld Model

Conduction electrons are described as a free quantum gas. Two electrons (spin-up and spin down) can occupy each of the dn states per unit wave vector.

*

22

2m

kE

The density of states between k and k+dk in a volume V is:

2

2

2dkVk

dn

Page 8: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

VkdkVk

N F

kF

2

3

02

2

322

m

kE F

F 2

22

m

kv F

F

n kF vF EF

4.6·1022cm-3 1.12Å-1 1.23·108cm/s 4.74eV

For lithium (a, lattice constant=3.49Å):

2

3

3Fk

V

Nn

Putting 2 electrons (spin up/spin down) into each of the dn states at T=0:

Electron density = valence electrons per unit cell / unit cell volume

Page 9: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

Transport in free electron metalsTransport in free electron metals

F

B

E

Tknf )(

Copyright Stuart Lindsay 2008

Transport involves only a fraction of carriers:

At 300K in Li:

0050744

0250.

.

.)n(f

At low applied electric fields, main source of excitation is thermal.

Page 10: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

Modified Drude TheoryModified Drude Theory

F

B

E

Tk

m

ne

21

F

BBV E

TknkC

2

3

From Debye theory:

Copyright Stuart Lindsay 2008

3TCV

Page 11: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

Electrons in crystals: Bloch’s theoremElectrons in crystals: Bloch’s theorem

)()( R rUrU R: lattice translation vector

akmax

2

)( exp ,, rRkRr knkn i Bloch’s theoremBloch’s theorem

All the properties of an infinite crystal can be described in terms of the basic symmetries and properties of a unit cell of the lattice.

Crystal momentum

Wavelenghts less than a lattice constant are not physically meaningful. Measurable quantities must always have the periodicity of the lattice. For an infinite crystal: k=0.

Page 12: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

aa

k

First Brillouin zone

-k and k directions are equivalent (reduced zone scheme)

n

sT narikna )(exp

Trial Bloch states for an 1D lattice:

From I order Perturbation Theory:

Top

Tsk UEE

Electrons in the lattice

Single electrons in isolated atoms

Interactionhamiltonian

Page 13: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

Band structureBand structure

ikaexp)ar(U

ikaexp)ar(UUEE

sop

s

sop

ssop

ssk

Nearest neighbors approximation (n=±1):

0 sop

ss UE )( arU sop

s

On-site energy Hopping matrix element

Copyright Stuart Lindsay 2008

Page 14: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

0)(

dk

kdE

Negative effective mass!

2

2

2

)(1

k

kEm

*

22

2m

kE

Free electronsElectrons in a periodic potential

Copyright Stuart Lindsay 2008

kaEk cos20

Page 15: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

dk

)k(dE

dk

dvg

1

group velocity

At ka =±π the group velocity falls to zero: E=ε0.

ak

22

Bragg diffraction condition

The combination of forward and backward (reflected) wave results in a standing wave: the electron does not propagate at these values of k.

The flattening of the function E(k) causes an increase in the density of states near ka=±π.

Page 16: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

Extended and reduced Brillouin zones

Extended Zone Reduced Zone

Copyright Stuart Lindsay 2008

Page 17: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

Band structure and electronic propertiesBand structure and electronic properties

• MetalsMetals : EF lies inside an allowed band (1 electron/unit cell)

• InsulatorsInsulators : The Fermi level lies at the top of a band (full band). Large gap between bands.

• SemiconductorSemiconductor: Full band (valence band).

- Dope with free electrons in the conduction band (e.g., P, As):

m*>0 (n type donors, negative carriers)

- Dope to take electrons from valence band (e.g., B, Ga):

m*<0, “positive” carriers (holes) (p type donors)

TkU B2

TkU B2

(Dielectric breakdown for large electric fields)

Page 18: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

Why elements with 2 electrons/unit cell are most metals ?

Cubic lattice in the reciprocal space

A cube of side 2π/a

Density of states(free electron model)

A Fermi sphere of radius kF

1° BZ completely filled, 2° BZ partially filled.

Partially filled

Page 19: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

Electrons in a quantum point contactElectrons in a quantum point contact

A bias V, applied across the two electrodes, will shift the Fermi levels of one relative to the other by an amount eV.The net current will be proportional to the number of states in this energy range.

Filled states

Page 20: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

At what size quantum effects dominate?

Upper limit:the size of the nanostructure approaches the electron mean free path for scattering (tens to hundreds of nm at room temperature).

Lower limit: only one mode of transmission available in the channel, i.e. Fermi wavelength in diameter (2π/kf).

For lithium ≈ 6Å (atomic dimensions)

Page 21: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

Fermi Golden Rule can be applied also to the case of electrons that tunnel from one bulk electrode to another by means of a small connecting constriction.

From Perturbation Theory:

)(ˆ2),(

2

kkm EHkmP

Density of states close to Ek

Probability of transition from m to k

Page 22: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

The Landauer ResistanceThe Landauer Resistance

vi ne Intensity of current per unit area

eVdE

dk

dk

dneV

dE

dnn

2

1

dk

dn

In 1D the distance between allowed k points is 2π/L.

Per unit length:

no. of states in the energy range dEper unit energy (eV)

dk

dEvv g

1 group velocity

eVv

ng

1

2

1

Page 23: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

For N=1:S

h

eG 5.77

2 2

0

k.G

RL 9121

0

Landauer resistance:

N = allowed quantum states in the channelThe factor 2 accounts for the two allowed spin states.

Vh

eNeveV

vNnevi g

gg

221

2

12

The Landauer resistance is independent of the material lying between the source and the sink of electrons. It is a fundamental constant associated to quantum transport.

Page 24: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

Landauer resistance is NOT a resistance in the ohmic sense:no power is dissipated in the quantum channel!

It reflects how the probability of transmission changes as the applied voltage is changed.

If the restriction is smaller than the scattering length of the electrons, it cannot be described as a resistance in the ‘Ohm’s law” sense. Dissipation requires scattering.

This occurs in bulk electrodes, but not in the nanochannel!

This is the reason why the high current densities in the STM (109 A/m2) do not damage the sample.

Page 25: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

If the source and sink of electrons are connected by N channels (N different electronic wavefunctions can occupy the gap), the resistance of the gap is Rg is:

22

11

e

h

NR

NR Lg

The resistance of a tunnel junction of gap L is:

L.exp.L.expRR L 021912021

Φ = V0-E = workfunction of the metal

Page 26: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

A. Propagation of quantum modes in a very narrow channel

Break junctionsBreak junctions

B. Landauer steps in the conductance of a gold break junction.

As the wire narrows down to dimensions of a few Fermi wavelengths, quantum jumps are observed in the current.

Page 27: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

An exact conductance can be calculated from the Landauer-Buttiker equation:

222

j,iijT

h

eG

Tij = matrix elements that connect electronic states i on one side of the junction to states j on the other side.

Page 28: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

The Coulomb BlockadeThe Coulomb Blockade

Bulk electrode

nm-sized conducting islandET can occur by hopping or

by resonant tunneling through the island

Single Electron TransistorSingle Electron TransistorA gate electrode applied to the island can alter its potential to overcame the blockade

Page 29: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

Two possible Electron Transfer mechanisms:

Resonant tunnelingResonant tunneling: electron tunnels straight through the whole structure.

Hopping mechanismHopping mechanism: electron hops on the center particle and then hops on the other electrode.

The hopping mechanism (negligible tunneling regime) is very sensitive to the potential of the center particle because the charging effect of a small particle for the transfer of one electron can be quite significant.

If the charging energy is greater than the thermal energy available, further hopping is inhibited (Coulomb blockadeCoulomb blockade).

Page 30: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

a

eV

08

When the applied bias exceeds the Coulomb blockade barrier current can flow again.

→ Coulomb staircaseCoulomb staircase

The voltage required to charge a spherical island of radius a is given by:

Taking: ε=1 and ε0=8.85·10-12 F·m-1 one obtains:

ΔV = 0.7 V for a = 1nm

ΔV=7 mV for a = 100nm

Condition for Coulomb blockade is that the electron localizes on the quantum dot (negligible tunneling).

Page 31: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

• Hanna-Tinkham theoryHanna-Tinkham theory

Electric circuit model of the two junctions Coulomb blockade experiment.

n

n

n

n

)n(e)n(e)V(I 1122

)n( Normalized distribution of charges on the central particle

j Thermally activated hopping rates between the particle and the left or right electrodes,

Page 32: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

I-V curves from nanoscale double junctions experiments. dots: experimental points; lines: Coulomb blockade theory.

Q0 = residual floating charge on the island

Hanna and Thinkham, Phys. Rev. B, 1991.

Page 33: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

Single Electron TransistorSingle Electron Transistor

Electrodes: n-GaASIsland: n-GaAs circular quantum dotInsulator: AlGaAs

A finite source-drain voltage (Vsd) opens a window of potential for tunneling via the quantum dot.

= an isolated metal particle coupled by tunnel junctions to two microscopic electrodes.The isolated metal particle is coupled with a gate electrode that allows to control the potential of the metal particle independently.

Page 34: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

White areas (Coulomb staircase):

0sddV

dI

Gate bias for level at Vsd = 0

A 3D-plot: dI/dVsd (z-axis) as a function of the source-drain potential applied between the electrodes and the gate potential applied to the quantum dot.

Red areas: SET is on

maxdV

dI

sd

at Coulomb steps

Page 35: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

SET as a micromechanical sensorSET as a micromechanical sensor

R.G. Knobel and A.N. Cleland, Nature 2003 424, 291

A sensor with single-electron sensitivity!

An electrochemical sensor based on the capacitive coupling of a vibrating beam to the gate of a SET.

Page 36: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

Resonant tunnelingResonant tunneling

central particlediameter = 2R

L Tunneling rate from the leftR

Tunneling rate to the right

Electrons incident from the left face a barrier of height V0 containing a localized state at energy E0.

Page 37: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

220

2

)()(

4

RL

RL

EEh

eG

Zero-bias

conductance

(2 counts both spin channels)

G = 0.5 G0 when L=R and E = E0

At resonance (E=E0), the localized state is acting like a metallic channel that connects the left and right electrodes.

If the tunneling rates are small enough, charge accumulation on the localized state becomes significant, resulting in Coulomb blockade.

The Coulomb blockade requires that the tunneling resistance of the contacts to the central particle exceeds twice the Landauer resistance (i.e. h/e2).

Page 38: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

Transmission vs. energy for a tight-binding model of a resonant tunneling through a molecule bound into a gap in a 1D wire.

A) 2 states in the conduction band, L=R C) 2 states, L=4R

B) 1 state Dashed lines: electronic transmission expected through the gap if no molecule is present between the two electrodes.

In case of strong coupling betwwen the electrodes and the quantum dot, tunneling predominates and the whole system must be tretaed quantum mechanically.

Page 39: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

Time development of the charge density for a wave packet incident from the left on a pair of barriers containing a localized resonant state. Electron is modeled as a Gaussian wave packet launched from the left.

a-d:small barriersstrong coupling

e-h:large barriersweak coupling

Page 40: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

Localization in disordered systemsLocalization in disordered systemsThe impact of disorder on electron transport becomes more significant in nanometric systems.

• Temperature dependence of resistivity in nanostructures

Charge density distribution calculated for electrons in random potentials.

W/V = width of the potential distribution in relation to the mean value of the potential.

For W/V=8 the electrons are almost completely localized.

Page 41: Electrons in Nanostructures The electronic properties of bulk materials are dominated by electron scattering. Electrons travel at a drift velocity depending

Peierl’s distortion: Peierl’s distortion: observed in linear conductive polymers

Localization in nanometric structuresLocalization in nanometric structures

This distortion results in halving of the Brillouin zone in wave vector space because the real space lattice is now doubled in size. metallic (half-filled band) to insulator (full band) transition