Meso 1 Intro 2011

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Mesoscopic Physics

Victor Moshchalkov (coordinator)André Stesmans - Michel Houssa

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I. Introduction to the mesoscopic scale

II. Electronic transport in mesoscopic systems

III. Graphene and carbon nanotubes

IV. Magnetism and superconductivity at mesoscopic scale

OUTLINE

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I. Introduction to the mesoscopic scale

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Meso (greek) means ‘’in-between ’’

Mesoscopic scale lies in-betweenmicroscopic and macroscopic world

Number of atoms100 102 104 106 108 1010 1012

Microscopic

Quantum mechanics

Macroscopic

Classical mechanics

MesoscopicAtom pendulum

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A typical example : electrical resistance

+ -

I

V

L

dW

The electrical resistance R of a (macroscopic) 3D metallic bar of lengthL and cross section A=W*d is given by

d W

L R ρ =

ρ is the electrical resistivity of the material (a material’s propertyindependent of its dimensions).

In the 2D-limit (e.g. for d 0), the resistance is given by

W

L R ρ =

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The resistance is thus expected to scale with L and W -1.In this classical picture, the resistance can take any value, depending on the material’s resistivity

In mesoscopic systems, the resistance is quantized,

G ( e

2 / h )

-2.2 -2.0 -1.8 -1.6 -1.4V g

Quantization of R isa consequence of thequantization of theelectron energy levelsin the system

22e

hn R =

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Mesoscopic scale is thus related to the size of a systemwhere quantum mechanical effects can be observed

In the case of electrical transport, the classical expression of the resistance breaks-down when the size of the systembecomes small compared to 3 characteristics lengths

- the electron wavelength

- the electron mean free path

- the phase coherence length

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Electron wavelength

In quantum mechanics, electrons are described by their wave function ψ(r ,t)

k: electron wave-vector

ω: pulsation

λ π /2=k

t ir k i e Aet r ω ψ −

=.),(

/ E =ω

λ is the electron wavelength (also called de Broglie wavelength)

Typical values of λ : 1-5 Å in metals10-100 nm in semiconductors

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Electron mean free path

The electron mean-free path l corresponds to the typicaldistance traveled by an electron, subjected to an electric field,before encountering a collision or scattering event, leading tothe change in its momentum p (or velocity v)

e

defect

e

vm p e= vm p e

′=

′

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The electron mean free path is strongly dependent on temperature.In the Drude model, l is related to the electron mobility µ through

the expressionn: electron density

l~100 µ m at 10 K

l~1 µ m at 300 K

e

nl

π µ 2=

Electron mobility and density in a 2DEG

GaAs/AlGaAs heterostructure

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Phase-coherence length

The phase-coherence length L ϕ is related to the wave nature of the

electrons. It represents the characteristic length the electrons can travelbefore their phase is changed or “destroyed”, through a collision.Consider an electron beam which is splitted into 2 paths, and thenrecombined. In a perfect crystal, the 2 paths would be identical, andinterference is constructive.

If impurities are present in the 2 arms of the loop, or if electrons can bescattered by phonons, or other electrons, the interference of the 2 outgoingelectron waves may be (partly) destructive, leading to the loss of the initialphase of the electrons (phase randomizing event).

Path 1

Path 2

τφ: average time by which theamplitude of the electron wave isreduced by exp(-t/ τϕ) after time t

=

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Such phase relaxation effects have been observed e.g. in ring-shapedresistors. The temperature dependence of the inverse of the phase

relaxation time τϕ is shown below, for the case of a GaAs sample.

Applying an AC magnetic field to thestructure allows one to obtain thephase of the electrons

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The phase coherence length can be calculated as

This equation is valid if the phase relaxation time is of the same order or shorter than the momentum relaxation time, τ

ϕ≤ τm. In this case, the

motion of electrons over a phase-relaxation time is nearly ballistic. This isusually the case in high-mobility semiconductors. Typical values for GaAsare few hundreds of nanometers.

In low mobility semiconductors (and also polycrystalline metal layers),momentum can relax much faster than the electron phase. In this case,electrons experience a diffusive motion during τ

ϕ, and the phase

relaxation length is then related to the diffusion coefficient D of thematerial

ϕ ϕ τ D L =

ϕ ϕ τ F V L =

wheree

T k D B

µ = (Einstein relation)

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Examples of mesoscopic systems

Ag cluster

Atomic clusters

Graphene-based devicesIII-V HEMT

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Example: atomic clusters

Aggregate of few to 105

atoms, also called nanoparticles

Molecule

Bulk material

Cluster

Number of atoms

106

1-10

TEM picture of a ~40 nmdiameter silver cluster

Typical cluster sizes: a few nm to few hundreds of nm

Ag cluster

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Metal-insulator transition in metallic clusters

A "macroscopic" metal is a very good conductor of electricity,but what about a cluster with a finite number of atoms?

CLUSTERS BULK METAL ATOMS &MOLECULES

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kT defines an energy to be compared with the energy gap δ of a particle (Kubo gap)

Approximate energy spacing : δ ≈ E F

n

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Electronic density of states of Hg clusters

Hg:(Xe)4f 145d106s2

s-states

p-states

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Work function/electron affinity of Hg clusters

“Low” workfunction, typical of a metal

“High” workfunction, typical of an insulator

Metal

EF

WF

Insulator

CB

VB

WF

Vacuum level

W o r k

f u n c t i o n / e l e c t r o n a f f

i n i t y ( e V )

Cluster size-induced metal/insulator transition

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Experimental realization of nano-scaled structures a. Top-down approach – Photolithography and etching

Photolithography consists in transfering a pattern, drawn on a mask,on a layer or a substrate. A typical lithography process used for patterning a metal gate on an oxide layer is illustrated below

Semiconductor

MetalOxide

1)

Photo-resist

Light (UV)exposure

Mask

Deposition of a photoresistive layer

2)

Exposure of the photoresist through a mask

Etching of the resist exposed to light

3)

Etching of the (unprotected) Al layer

4)

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Resist removal

5)

The photo-resist is a material (polymer) which undergoes a chemicaltransformation when exposed to light. The exposed part of the resistcan then be dissolved in an appropriate solvent (developer).

Patterning of MOS structures

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In order to pattern nanostructures (with typical sizes of few tens of nanometers), the projection lithographic process is used. In this

case, a system of lenses is used to project the image of the maskon the layer to pattern. The minimal size l m of the pattern that can betransferred on the layer is related to the numerical aperture of theoptical system and the wavelength of the light source

Reticle Optical system Focal plane

Layer

NAl m

λ ≈

For a lens, NA=sinθ

Decreasing wavelength allows to pattern smaller structures

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0

100

200

300

400

500

600

0 100 200 300 400 500

M i n i m u m

f e a

t u r e s

i z e

( n m )

Wavelength (nm)

UV light

X-rays

Potential of X-rays for patterning nm sized structures …

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During the fabrication of nanostructures, a photolithography step isusually followed by an etching step. Wet etching is based on thechemical attack (dissolution) of the layer to pattern in a solution.

Material to etch Chemicals Etching rate(µ m/min)

Si

SiO 2

Si 3N 4

Al

1 ml HF5 ml HNO 3

2 ml CH 3 COOH

28 ml HF170 ml H 2O113 g NH 4 F

H 3PO 4

1 ml HNO 3 4 ml CH 3 COOH

4ml H 3PO 4 1 ml H 2 O

7.4

0.1

0.01

0.035

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A limitation of the wet etching process lies in the fact that it is anisotropic process, i.e. the material is etched in each directionswith the same etching rate, i.e.

where v l and vv represents the etching rate along the lateral andvertical directions, respectively. If Af ≈ 1 (vv>>vl), etching isanisotropic

01 ≈−=

v

l f v

v A

Isotropic etching Anisotropic etching

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Isotropic etching of a layer leads to the partial loss of the resolution of the structure to be patterned on the layer, called over-etching. This

effect has to be avoided when structures of few tens of nm have to befabricated.

over-etching

Resist Resist

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Over-etching can be avoided by using a dry etching process, whichconsists in exposing the layer to be patterned to a plasma, containingchemically active ions. The plasma is generated from a gas, which isionized by a high electric field (usually an AC electric field pulsed atradio frequencies).

Dry etching of the layer then proceeds from a chemical component(due to the active ions like F - or Cl-) and a physical component, due to

the ablation of the material induced by its bombardement with thehigh energy ions.

Layer to be patterned

Substrate holder under bias

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The chemical component is isotropic and selective towards differentmaterials, while the physical component is anisotropic and non or weakly selective. Proper adjustment of the properties of the plasma(chemical content, pressure, energy of the ions impacting on thesurface) allows to maximize selectivity and anisotropy of the etchingprocess.

As an example, using CF 4 and O 2 enables the selective etching of Siwith respect to SiO 2 with a selectivity ratio of 12:1.

Material to etch Chemicals

SiSiO 2 Si 3 N 4

Al, Ti, WResist

CF 4 +O 2 , CCl 4 , Cl 2 CF 4 +H 2 , CHF 3 +O 2

CF 4 +O 2 CCl 4 , SiCl 4 , BCl 3 +Cl 2

O 2

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b. Bottom-up approach – Self-assembly

Self-assembly corresponds to the spontaneous organization of pre-existing components (atoms or molecules) by means of electrostatic or hydrophobic forces, this self-organization processallowing to reduce the energy of the system.

Starting from atoms/molecules to build up a functional structure

or device, so called ‘’bottom-up approach’’. Inspired from biology: DNA

Adenine

Cytosine

Guanine

Thymine

Bonding through

hydrogen bonds

DNA molecule

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A typical example of self-assembly: alkanethiols on a gold surface

The self-assembly of alkanethiols on Au substrates originates fromthe strong Au-S bonding (~1.7 eV), which follows from thedissociation of the S-H group on the surface:

Alkanethiol: an organic ‘’chain’’, terminated by an S-H (functional) group

R (C,H)

SH

22

1 H AuS R Au H S R +−−→+−−

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The process then leads to the formation of an ordered monolayer of R-S on the Au surface, due to the tetrahedral arrangement of

the S-H bond on the surface, as well as the (weak) Van der Waalsinteractions between the alkanethiols molecules.

30°

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Growth of carbon nanotubes (CNT) from catalysts

CNT grown onnanoparticlesFe, Co, Ni

nanoparticles

Chemical vapor deposition (CVD) of CNT, using Fe, Co or Ninanoparticles as catalysts. Typical carbon source is C 2H2 and thegrowth process takes place at about 1000 °C.

10 12 – 10 13 CNTs/cm 2

CNT on Fe nanoparticles

10 12 – 10 13 CNTs/cm 210 12 – 10 13 CNTs/cm 2

CNT on Fe nanoparticles

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Combining top-down and bottom-up approaches: CNT interconnects

Via level

catalysts at bottom of via

Trench level

catalysts at side wall

Replacement of Cu by CNTin nanoelectronic devices?

Bottom-up:growth of CNT

Top-down:

definition of vias and trencheson dielectric layers

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References

Y. ImryIntroduction to mesoscopic physics(Oxford University Press, 2002).

C.P. Poole and F.J. Owens,Introduction to Nanotechnology (Wiley, 2003).

M. Wilson, K. Kannangara, G. Smith, M. Simmons, and B. Raguse,Nanotechnology – Basic science and emerging technologies

(CRC Press, 2002).

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