Handout 27

1D and 0D Nanostructures: Quantum Wires and Quantum Dots

In this lecture you will learn:

• Semiconductor quantum wires and dots • 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 waveguide confinement

SEM of 20 nm diameter GaAs

A (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 Ec1k  Ec1  2me y Outside: 2 2 x   k Ec2k  Ec2  z 2me

Inside:  2 2  ˆ       Ec1 i 1r  E 1 r   Ec1   1r  E 1 r  2me  Outside:  2 2  ˆ       Ec2 i 2 r  E 2 r   Ec2   2r  E 2 r  2me  Assumed solutions: Inside: Outside:  i kz z  i kz z 1r  A f1x, y e 2r   A f2 x, y  e

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Semiconductor Quantum Wires Inside: Plug in the assumed solution: Ec2 Ec1  22  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  f1x, 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:

f1x, y boundary  f2x,y boundary nˆ is the unit vector 1 1 normal to the boundary ˆ ˆ f1x,y .n  f2x,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 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 Ec1Ep   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 )

CdSe

CdTe

InAs quantum dots (MBE)

TEM of a PbS

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 Ec1k  Ec1  2me Outside: 2 2   k Ec2 k  Ec2  2me

Inside:  2 2  ˆ       Ec1 i 1r  E 1 r   Ec1   1r  E 1 r  2me  Outside:  2 2  ˆ       Ec2 i 2r  E 2 r   Ec2   2r  E 2 r  2me 

Assumed solutions: Inside: Outside:   1r   A f1x, y,z 2r   A f2x, y,z

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

5 Semiconductor Quantum Dots Boundary conditions at the inside-outside boundary:

f1x, y,z boundary  f2x, y,z boundary nˆ is the unit vector 1 1 normal to the boundary ˆ ˆ f1x, y,z .n  f2x, 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

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