Electronic Transport in Nanostructures

Kashif SabeehQuaid-i-Azam University

Introduction

Nanofabrication has made it possible to study condensed matter systems at an unprecedented small scale.Driven primarily by the search for ever smaller devices, submicron nanodevices are under intense study.An important feature of these devices is that there behaviour is governed by Quantum Mechanics.

Mesoscopic Regime

Mesoscopic Scale

Modern electronic devices belong to the mesoscopic scale.The development of modern microelectronics industry is made possible by IC technology: The integration of large number of transistors into densely packed integrated circuits.

Fabricating MesoscopicConductors

Progress relies on initially turning semiconductors into metals by doping.It is necessary to separate dopants and charge carriers.Realization of a Two-Dimensional Electron Gas (2DEG) in semiconductor heterostructures.

Quantum Confined Nanostructures

Quantum dotsQuantum wiresTwo-dimensional electron gas systems

Semiconducting Quantum Dots

Quantum Wires

Nano Scale Building Blocks

Two-Dimensional Electron Gas

Novel Phenomena in TransportQuantized ConductanceQuantum Interference Effects:Aharonov-Bohm EffectWeak LocalizationCoulomb BlockadeQuantum Hall Effects

Characteristic Length Scales

The de Broglie wavelength: Related to the kinetic energy of the electrons.The mean free path: Distance an electron travels before its initial momentum is destroyed.The phase relaxation length: Distance an electron travels before its initial phase is destroyed.These length scales vary widely from one material to another and are also affected by temperature.When the dimensions of a conductor are larger than these lengths it exhibits Ohmic behavior.

Ohm’s Law

At the heart of electrical conductance is the Ohm’s Law V=IR.Historically, the discovery of electron by Sir J. J. Thomson offered an obvious mechanism for conduction in metals.The corresponding model, Drude model of electrical conduction, was presented by Paul Drude.

Question: How small can we make the dimensions, W and L, before this ohmic behavior breaks down?Conductors with these dimensions are mesoscopic.

Experimental Realization of Conductance Quantization

Bert van Wees experiments on quantum point contacts

Conductance from TransmissionMesoscopic transportCurrent through a conductor is expressed in terms of the probability that an electron can transmit through itIntuitively appealingLandauer raised subtle questions related to transport in ballistic conductors

Landauer Formula

N: Number of transverse modes or occupied channels.T: Average probability that an electron injected at one end of the conductor will transmit to the other end.

Landauer FormulaFor a ballistic conductor T=1, it says that the conductance is

Universal as it is given in terms of conductance quantum which only contains universal constants.

It is quantized as it scales with the integer number of occupied quantum channels.

Quantum Interference Effects

Coherent Backscattering and Weak Localization

Hamiltonian is invariant under time reversalConsider paths which start and finish at one point (chosen as origin).Classically: Return probability is the sum of the probabilities of propagating along each path separatelyQuantum Mechanically: The return amplitude is the sum of the amplitudes of different paths.Interference effects

Coulomb Blockade: Charge Quantization and Charging Energy

A many body phenomena where electron-electron interactions have to be considered.Related to quantization of charge.Realized in systems where a central region is weakly coupled to the leads. An electron that tunnels between leads through a small conductor charges itAt low T, as the electrostatic charging energy Ec~e^2/2C<<kT, thermally activated tunneling through the conductor is prevented leading to Coulomb blockade.The small conductor may be a metallic island, a quantum dot and even a single molecule.

Single electron transistor (SET)

Coulomb blockade regime:

Fractional Quantum Hall Effect

Graphene Nanoelectronics

Scanning electron micrograph of a graphene single electron transisitor

Klein tunneling through a time-periodic potential

Magnetotransport in Modulated Graphene

Conclusions

Electronic transport in nanostructures is a fascinating area of research.Close relationship between experiments and new physics.Transport in Graphene is the new frontier.