1ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityHandout 271D and 0D Nanostructures: Semiconductor Quantum Wires and Quantum DotsIn this lecture you will learn:• Semiconductor quantum wires and dots• Density of states in semiconductor quantum wires and dotsCharles H. Henry (1937-)ECE 407 – Spring 2009 – Farhan Rana – Cornell University1D Nanostructures: Semiconductor Quantum WiresSEM of 20 nm diameter GaAs nanowiresGaAs/AlGaAs quantum wires grown by electron waveguide confinementA carbon nanotube (rolled up graphene):2ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversitySemiconductor Quantum Wires12cccEEExyzEc2Ec1Inside:eccmkEkE22211Outside:eccmkEkE22222Inside:   rErmErEriEecc112211112ˆOutside:   rErmErEriEecc222222222ˆInside:zkizeyxfAr ,11Outside:zkizeyxfAr ,22Assumed solutions:1cE2cEECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityxyzxyzEc2Ec1Semiconductor Quantum Wires   yxfmkEEyxfymxmyxfEyxfymxmmkEeyxfEeyxfmEezceeeeezczikzikeczz,2,22,,222,,2122112222221122222222111221Inside:Plug in the assumed solution:Outside: yxfmkEEyxfymxmezcee,2,2222222222222Boundary conditions at the inside-outside boundary: boundary2boundary1,, yxfyxf  boundary2boundary1ˆ.,1ˆ.,1nyxfmnyxfmeeis the unit vector normal to the boundarynˆ3ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversitySolve these with the boundary conditions to get for the energy of the confined states:........3,2,12,221 pmkEEkpEezpczcThe electron is free in the z-direction but its energy due to motion in the x-y plane is quantized and can take on only discrete set of valuesxyzxyzEc2Ec1Semiconductor Quantum Wireszk1cE11EEc21EEc31EEcEThe energy dispersion for electrons in the quantum wires can be plotted as shown:It consists of energy subbands (i.e. subbands of the conduction band)Electrons in each subband constitute a 1D Fermi gasECE 407 – Spring 2009 – Farhan Rana – Cornell UniversitySemiconductor Quantum Wires: Density of StatesSuppose, given a Fermi level position Ef , we need to find the electron density:We can add the electron present in each subband as follows: pfzczEkpEfdkn ,22fEIf we want to write the above as:fQWEEEfEgdEnc1Then the question is what is the density of states gQW(E ) ?zk1cE11EEc21EEc31EEcE4ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityStart from:ezpczcmkEEkpE2,221And convert the k-space integral to energy space:fppcpceEpfpceEEEEfEEEEEEmdEEEfEEEmdEncpc11221222 211This implies:EgQW1cE11EEc21EEc31EEcSemiconductor Quantum Wires: Density of StatesfEzk1cE11EEc21EEc31EEcEpfzczEkpEfdkn ,22pcppceQWEEEEEEmEg1122 2ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversitySemiconductor Quantum Wire LasersGaAs/AlGaAs quantum wires grown by electron waveguide confinementmetalA Ridge Waveguide Laser Structure5ECE 407 – Spring 2009 – Farhan Rana – Cornell University0D Nanostructures: Semiconductor Quantum DotsTEM of a PbS quantum dotCore-shell colloidal quantum dots (Mostly II-VI semiconductors)CdTeCdSeGaAs substrateInAs quantum dots (MBE)ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversitySemiconductor Quantum Dots12cccEEEEc2Inside:eccmkEkE22211Outside:eccmkEkE22222Inside:   rErmErEriEecc112211112ˆOutside:   rErmErEriEecc222222222ˆInside:zyxfAr ,,11Outside:zyxfAr ,,22Assumed solutions:Ec11cE2cE6ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityBoundary conditions at the inside-outside boundary: boundary2boundary1,,,, zyxfzyxf  boundary2boundary1ˆ.,,1ˆ.,,1nzyxfmnzyxfmeeis the unit vector normal to the boundarynˆSemiconductor Quantum DotsEc2Ec1Solve these with the boundary conditions to get for the energy of the confined states:........3,2,11 pEEpEpccThe electron is not free in any direction and its energy due to motion is quantized and can take on only discrete set of valuesIn the limit Ec ∞ the lowest energy level value for a spherical dot of radius R is:2212RmEeECE 407 – Spring 2009 – Farhan Rana – Cornell UniversitySemiconductor Quantum Dots: Density of StatesSuppose, given a Fermi level position Ef , we need to find the electron number N:We can add the electron present in each level as follows: pfcEpEfN2If we want to write the above as:fQDEEEfEgdENc1Then the question is what is the density of states gQW(E ) ?Ec2Ec1pcQDpEEEg2Because the dot is such a small system, at many times concept of a Fermi level may not even be appropriate!!EgQD1cE11EEc21EEc31EEc7ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityA ridge waveguide quantum dot laser structuremetalSemiconductor Quantum Dot Lasers (III-V Materials)non-radiativerecombinationelectronsN-dopedphotonstimulated andspontaneousemissionholesP-doped• Only 2 electrons can occupy a single quantum dot energy level in the conduction band• Only 2 holes can occupy a single quantum dot energy level in the valence band Some advantages of 0D quantum dots for laser applications:• Ultralow laser threshold currents due to reduced density of states• High speed laser current modulation due to large differential gain• Small wavelength chirp