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Semiconductor Quantum Wires and Quantum Dots Handout 27 1D and 0D Nanostructures: Semiconductor Quantum Wires and Quantum Dots In this lecture you will learn: • Semiconductor quantum wires and dots • Density of states in semiconductor quantum wires and dots Charles H. Henry (1937-) ECE 407 – Spring 2009 – Farhan Rana – Cornell University 1D Nanostructures: Semiconductor Quantum Wires GaAs/AlGaAs quantum wires grown by electron waveguide confinement SEM of 20 nm diameter GaAs nanowires A carbon nanotube (rolled up graphene): ECE 407 – Spring 2009 – Farhan Rana – Cornell University 1 Semiconductor Quantum Wires Ec2 Ec Ec2 Ec1 Ec1 Inside: E E 2 2 c2 c1 k Ec1k Ec1 2me y Outside: 2 2 x k Ec2k Ec2 z 2me Inside: 2 2 ˆ Ec1 i 1r E 1 r Ec1 1r E 1 r 2me Outside: 2 2 ˆ Ec2 i 2 r E 2 r Ec2 2r E 2 r 2me Assumed solutions: Inside: Outside: i kz z i kz z 1r A f1x, y e 2r A f2 x, y e ECE 407 – Spring 2009 – Farhan Rana – Cornell University Semiconductor Quantum Wires Inside: Plug in the assumed solution: Ec2 Ec1 22 E f x,y eikzz E f x, y eikzz y c1 1 1 x 2me z 2 2 2 2 2 2 kz Ec1 2 2 f1x, y E f1 x, y 2me 2me x 2me y 2 2 2 2 2k2 f x, y E E z f x, y 2 2 1 c1 1 2me x 2me y 2me Outside: 2 2 2 2 2k2 f x, y E E z f x, y 2 2 2 c2 2 2me x 2me y 2me Boundary conditions at the inside-outside boundary: f1x, y boundary f2x,y boundary nˆ is the unit vector 1 1 normal to the boundary ˆ ˆ f1x,y .n f2x,y .n boundary me boundary me ECE 407 – Spring 2009 – Farhan Rana – Cornell University 2 Semiconductor Quantum Wires Solve these with the boundary conditions to get for the energy of the confined states: Ec2 Ec1 2k 2 E p,k E E z p 1,2,3........ y c z c1 p x 2me z 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 values E The energy dispersion for electrons in the quantum Ec1 E3 wires can be plotted as shown: Ec1 E2 It consists of energy subbands (i.e. subbands of the Ec1 E1 conduction band) E Electrons in each subband constitute a 1D Fermi gas c1 kz ECE 407 – Spring 2009 – Farhan Rana – Cornell University Semiconductor Quantum Wires: Density of States Suppose, given a Fermi level position Ef , we need to find the electron density: E We can add the electron present in each subband as follows: E E dkz c1 3 n 2 f E p,k E Ef c z f Ec1 E2 p 2 Ec1 E1 If we want to write the above as: Ec1 n dE gQW E f E Ef kz Ec1 Then the question is what is the density of states gQW(E ) ? ECE 407 – Spring 2009 – Farhan Rana – Cornell University 3 Semiconductor Quantum Wires: Density of States 2 2 kz E Ec p,kz Ec1 Ep 2me Start from: E E E c1 3 f E E dkz c1 2 n 2 f Ec p,kz Ef p 2 Ec1 E1 And convert the k-space integral to energy space: Ec1 2me n dE 2 2 f E Ef kz p Ec1Ep E Ec1 Ep 2me dE 2 2 E Ec1 Ep f E Ef Ec1 p E Ec1 Ep gQW E This implies: 2me gQW E 2 2 p E Ec1 Ep E Ec1 Ep Ec1 Ec1 E1 Ec1 E2 Ec1 E3 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Semiconductor Quantum Wire Lasers A Ridge Waveguide Laser Structure GaAs/AlGaAs quantum wires grown by electron waveguide confinement metal ECE 407 – Spring 2009 – Farhan Rana – Cornell University 4 0D Nanostructures: Semiconductor Quantum Dots Core-shell colloidal quantum dots (Mostly II-VI semiconductors) CdSe CdTe InAs quantum dots (MBE) TEM of a PbS quantum dot GaAs substrate ECE 407 – Spring 2009 – Farhan Rana – Cornell University Semiconductor Quantum Dots Ec2 Ec Ec2 Ec1 Ec1 E Inside: c2 2 2 E k c1 Ec1k Ec1 2me Outside: 2 2 k Ec2 k Ec2 2me Inside: 2 2 ˆ Ec1 i 1r E 1 r Ec1 1r E 1 r 2me Outside: 2 2 ˆ Ec2 i 2r E 2 r Ec2 2r E 2 r 2me Assumed solutions: Inside: Outside: 1r A f1x, y,z 2r A f2x, y,z ECE 407 – Spring 2009 – Farhan Rana – Cornell University 5 Semiconductor Quantum Dots Boundary conditions at the inside-outside boundary: f1x, y,z boundary f2x, y,z boundary nˆ is the unit vector 1 1 normal to the boundary ˆ ˆ f1x, y,z .n f2x, y,z .n boundary me boundary me Solve these with the boundary conditions to get for the energy of the confined states: Ec p Ec1 Ep p 1,2,3........ In the limit Ec ∞ the lowest energy level value for a spherical dot of radius R is: E 2 2 c2 E1 Ec1 2me R The electron is not free in any direction and its energy due to motion is quantized and can take on only discrete set of values ECE 407 – Spring 2009 – Farhan Rana – Cornell University Semiconductor Quantum Dots: Density of States Suppose, given a Fermi level position Ef , we need to find the electron number N: Ec2 E We can add the electron present in each level as follows: c1 N 2 f Ec p Ef p If we want to write the above as: gQD E N dE gQD E f E Ef Ec1 Then the question is what is the density of states gQW(E ) ? gQD E 2 E Ec p p Because the dot is such a small system, at Ec1 Ec1 E1 Ec1 E2 Ec1 E3 many times concept of a Fermi level may not even be appropriate!! ECE 407 – Spring 2009 – Farhan Rana – Cornell University 6 Semiconductor Quantum Dot Lasers (III-V Materials) electrons Some advantages of 0D quantum dots for stimulated laser applications: and spontaneous • Ultralow laser threshold currents due to emission reduced density of states non-radiative photon • High speed laser current modulation due recombination to large differential gain N-doped P-doped • Small wavelength chirp in direct current modulation • Ability to control emission wavelength via quantum size effect holes A ridge waveguide quantum dot metal laser structure • Only 2 electrons can occupy a single quantum dot energy level in the conduction band • Only 2 holes can occupy a single InGaAs quantum dot energy level in the InAs valence band Qdots ECE 407 – Spring 2009 – Farhan Rana – Cornell University Colloidal Quantum Dots: Wonders of Quantum Size Effect CdTe E photon 2 2 E 1 E c c Ec1 2me R photon CdSe 2 E v1 Ev 1 Ev 2m R h gQD E Photoluminescence from CdSe/ZnS Photoluminescence from (core-shell colloidal) quantum dots of CdTe/CdSe (core-shell different sizes (~2-6 nm) pumped with the colloidal) quantum dots of same laser different sizes ECE 407 – Spring 2009 – Farhan Rana – Cornell University 7 Quantum Dots: Biology Applications CdSe/ZnS quqntum dot coated with DHLA and functionalized with maltose binding protein (MBP) and Avidin Motion of quantum-dot-attached-RNA into cells monitored by the luminescence (the quantum dots used are CdSe (core) and ZnS (shell) ECE 407 – Spring 2009 – Farhan Rana – Cornell University Quantum Dots: Biology Applications Invitro microscopy of the binding of EGF to erbB1 erB1 bound to eGFP (enhanced green fluorescent protein) EGF (epidermal growth factor) bound to quantum dot Movie shows binding of EGF tagged with fluorescent quantum dots to erB1 tagged with the green fluorescent protein Nat. Biotechnol., 22, 198-203 (2004) CdSe Polymer coating ZnS Imaging of antibody (PSMA) coated quantum dots targeting cancer tumors cells Nat. Biotechnol., 22, 969 (2004) ECE 407 – Spring 2009 – Farhan Rana – Cornell University 8 Colloidal Quantum Dot Electrically Pumped LEDs eV Vacuum level 0 -1 -2 -3 -4 NiO 3 2 -5 WO ZnS ZnO 2 -6 CdSe ZnCdSe SnO ZnCdS -7 (ZTO) ZnSnO (ZTO) -8 Possible Possible hole Quantum electron dot injection Bulovic et. al. (2010) injection layers (QD) layers ECE 407 – Spring 2009 – Farhan Rana – Cornell University 9.
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