2. Length Scales and Low Dimensionality. Electron States and Quantum Confinement. Length Scales and Low Dimensionality

2. Length scales and low dimensionality. Electron states and quantum confinement.

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•Contents:

Length scales and low dimensionality.

• Introduction: Nanoscience and Mesoscopic Physics. • Dimensionality definitions. • Relevant length scales. • Examples of low dimensional systems. • Fabrication and exploring tools. • New phenomena and new applications.

2 • Introduction: Nanoscience and Mesoscopic Physics.

• MESO- In between an atom and bulk solids. Size below which a solid does no longer behave bulk-like. • Mesoscopic Physics...Physics of small condensed objects (a collection of atoms) • Often in the nanometer-size regime ! discipline of “Nanoscience”

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• Introduction: Nanoscience and Mesoscopic Physics.

Nanoscience and Nanotechnology

Why increasing interest for the nanoscale?

1 nm = 0,000000001 m

4 • Introduction: Nanoscience and Mesoscopic Physics.

Diameter of human hair

Diameter of red blood cell

Visible light wavelengths

Intel’s newest transistor

ADN

Diameter of DNA, nanotubes

Bohr diameter

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httpp://www.owlnet.rice.edu

• Dimensionality definitions I: Bond percolation (Microscopic scheme)

(Bottom-up)

•Based on the bonding.

•Strong covalent bond within regions of structure define the dimensionality unit and weak (e.g. Van der Waals) between units to produce the 3D structure overall.

6 • Dimensionality definitions

0D: Molecular P4Se3 1D: crystalline SiSe2

2D: crystalline Ge4Se3 3D:7 amorphous SiO2

• Dimensionality definitions I: Bond percolation (Microscopic scheme) (Bottom-up)

•Start by considering electrons in single atoms and small molecules. •Theories to treat electrons in nanostructures: Huckel theory “Tight-binding” Localized orbitals

•Chemistry.

•This point of view will be explored in the last chapter (C nanostructures)

8 • Dimensionality definitions

II: Length scales (More macroscopic scheme) (Top-down)

•Based on size dependence of a physical property, e.g. transport (electrons and also phonons involved).

•Reduced dimension if the dimension of the sample is lower than a characteristic length (e.g. mean free path for transport, Fermi wave-length for quantization or exciton Bohr radius in semiconductors).

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• Dimensionality definitions

L0 = λ , characteristic length

L >L,i=1,n (3 n)D system 0 i → −

0D: quantum dot

Lx,Ly,Lz

1D: quantum wire 2D: quantum well Lx,Ly

II: Length scales (More macroscopic scheme) (Top-down)

•Start from solid state physics. •Physics/electrical engineering. •Shows qualitative features. Not bad for many metals and doped semiconductors. •Approximations to treat electrons in nanostructures: • “Free electrons”-no external potential- • Independent electron approximation- ignores interactions. • Many-particle system can be modeled by starting from single particle case • Starting point: single particle states and energies (next lecture).

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• Relevant length scales

•A few relevant scale lengths:

proccess transistor (Texas Instruments).

12 S. Datta, “Electronic transport in mesoscopic systems”,1995 • Relevant length scales •Some characteristic lengths: •De Broglie wave length, Fermi wavelength:λ, λF 2π! 2π Related to kinetic energy of electrons λ = = p k 2π Fermi gas: characteristic momentum kF λF = → kF

One single filled band in 2DEG: λ = 2π/ns ,ns : sheet density Boltzamann gas: p = √2mkT !

•Mean free path:Lm

Initial momentum of electrons is destroyed

Length between collisions with impurities or phonons

Lm = vτt transport typical relaxation time velocity 13

• Relevant length scales •Some characteristic lengths:

•Phase-relaxation length : Lϕ Initial phase of electrons is destroyed Quantun mechanical: phase of the electron wave function

Lϕ = Dτϕ typical ! difusion time of D = (1/d)vLm constant elastic collisions dimensionality of electron gas •Thermal dephasing length : LT

Characteristic length of coherent propagation for two electrons If the energy difference between two electrons is ~kT, they travel almost coherently during time !/kT

LT = !D/kT ! 14 • Relevant length scales

•For example: Transport through a constriction, 3 different regimes:

•Wire dimensions: W, L Lm << W, L Lm •Mean free path: Lm

W < Lm

Lm >> W, L

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• Relevant length scales •For example:conductance quantization in a quantum point contact

By applying voltages to t h e p l a n a r g a t e electrode, the width of the wire is tuned.

At low T conductance is quantized in units of 2e2/h.

AFM surface topography of T. Heinzal, “Mesoscopic electronics in solid state nanostructures”, Wiley Ga AS microchip. Ballistic regime A small wire length 140nm, width 80 nm connects source and drain. Planar gate 30 nm below its surface. • Relevant length scales

•For example:mesoscopic ring used to study Abraronov-Bohm effect

L 100µm ϕ ∼ (low T)

A significant fraction of electrons traverse the ring phase coherently

From a 38 nm film of polycrystalline gold. Diameter 820 nm. Thickness of wires 40 nm

S. Washburn and R. A. Webb, Adv. Phys. 35, 375 (1986).

17 G. Fraser, “The New Physics for the 21th century”, Y. Imry, ch. 12

• Relevant length scales

•Summary of conditions required for a mesoscopic device

T. Heinzal, “Mesoscopic electronics in solid state nanostructures”, Wiley • Relevant length scales

•For example:Kondo mirage

Unusual phenomena due to the wave nature of electrons and their correlations around impurities.

Images of elliptical arrangements of atoms on a metallic surface, prepared and visualized with STM microscope.

Placing a magnetic impurity at a focal point the ellipse created a shadow in the other focus (“Kondo mirage”)

D. Eigler et al., IBM Almaden

http://www.almaden.ibm.com/almaden/media/image_mirage.html (This and more beatiful images)

• Examples of low dimensional systems

•Some quasi-two-dimensional systems:

G. Lehmann, P. Ziesche, “Electronic properties of metals, Esevier, 1990

MCBJ technique to produce metallic nanowires

E. Sheer et al., Phys. rev. Lett. 78, 3535 (1997) • Examples of low dimensional systems

•Peroskite-like high temperature superconductors

G. Lehmann, P. Ziesche, “Electronic properties of metals, Esevier, 1990

Superconductivity related to 2D character due lo weakly connected 2D sheets of Cu and O

• Examples of low dimensional systems •Some quasi-one-dimensional materials:

G. Lehmann, P. Ziesche, “Electronic properties of metals, Esevier, 1990 • Examples of low dimensional systems Carbon in all dimensions

sp2 Covalent C-C bonds within 'molecule' sp2

Variable sp pure sp2 hybridisation ! + " + - - pure sp3

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• Examples of low dimensional systems

•Semiconductor nanostructures starting from GaAs-AlGaAs heterostructures

•Diminishing dimensions...

•2D electron gas

E. Corcoran, TRENDS IN MATERIALS: DIMINISHING DIMENSIONS; November, 1990

24 • Examples of low dimensional systems

•Semiconductor nanostructures starting from GaAs-AlGaAs heterostrcutures

•Squeezing 2D electron gas...

QUANTUM WIRE

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• Examples of low dimensional systems

26 • Examples of low dimensional systems

•Also with Si (MOSFET)

Si technology

• Examples of low dimensional systems •Why GaAs?

C.W.J. Beenakker, H. van Houten, "Quantum Transport in Semiconductor Nanostructures", Solid State Physics 44, 1, 1991. http://arxiv.org/abs/cond-mat/0412664 28 • Examples of low dimensional systems

•OD systems, quantum dots or “artificial atoms”

•Clusters of metallic atoms grown from vapour-phase

condensation size increasing •Fullerenes •Synthetic nanocrystals :CdS, CdSe in glassy matrix, CuCl in NaCl crystals, Si, Ge... , Size control (~1nm-> ~200nm)

•Self-assembled QD’s

•QD’s produced from heterostructures and lithographic etching. applications in nanoelectronics and optoelectronics

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• Examples of low dimensional systems •OD systems, quantum dots or “artificial atoms”

• Synthetic nanoparticles interesting because of optical properties. • Reducing the size the gap changes, Higher fusion temperatures, estructural changes... e.g. the gap of CdSe can be tuned from red (1.7eV) to green (2.4 eV) when the particle diameter is reduced from 200 nm to 2 nm •Aplications: lasers, LEDS, biosensors.... • Fabrication and exploring tools

•Nanolithography •Atomic force microscopy •Scanning tunneling microscopy •Molecular beam epitaxy and other techniques for atomic-scale layer deposition of material. •Chemical sysntesis with different methods....

(Described in previouslecture)

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• New phenomena and new applications

•Laboratory for quantum phenomena: •quantum coherence, quantum confinement, tunel effect, electron-electron interactions.... •When we go dowm in dimension properties are not scalable: •new functional relations among magnitudes, oscillations of the physical magnitudes....

•New phenomena •quantum Hall effect, Coulomb blockade, breakdown of Ohm’s law, quantum size effects... •New operating principles and applications: one electron devices, molecular electronics, spintronics, nanophotonics. optoelectronic devices, quantum computing, bio-nano devices for aplications in biomedicine....

32 • New phenomena and new applications

•Scalability regimes:

Simulations of breaking of Na nanowires

Eduardo Ogando, Thesis 2004

• New phenomena and new applications

•General trends or signatures of low dimensionality

•Fermi surface topology for 3D (sphere), 2D (cylinder) and 1D (planes) electron gas

Fermi surface of a quasi-one -dimensional electron gas. Wavy planes due to weak coupling in real systems or

(More details in next lecture) • New phenomena and new applications

•General trends or signatures of low dimensionality

•Density of states

Eduardo Ogando, Thesis 2004 (More details in next lecture)

• New phenomena and new applications

•General trends or signatures of low dimensionality •Response function, susceptibility

Wave vector dependent response The response function of a 1D free function for 1”, 2D, and 3D electron gas electron gas at various temperature at T=0 K (Heeger, 1979) • New phenomena and new applications

•Response to magnetic fields (quantum Hall effect)

Measure the longitudinal (Rxx) and Hall resistance (Rxy) of a 2D electron gas as a Shubnikov-de Haas oscillations and function of the perpendicular magnetic the quantum Hall effect. field.

T=100mK

von Klitzing et al. 1982 G. Fraser, “The New Physics for the 21th century”, Y. Imry, ch. 12

• Summary

Write it yourself and send it to me (just to fill one slide)

38 • Take home exercises

(Be very concise)

•Bibliographic search: Peculiarities or surprises found for other low dimensional systems. Give paper reference where it is found, describe briefly the system (composition, size, tempertaure...) and the property studied.

•Find examples of systems behaving as 0D

•Why interest in AsGa? Compare properties of GaAs vs. Si

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