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ECE 407 – Spring 2009 – Farhan Rana – Cornell University

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

SEM of 20 nm diameter GaAs nanowires

GaAs/AlGaAs quantum wires grown by electron waveguide confinement

A carbon nanotube (rolled up graphene):

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ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Semiconductor Quantum Wires

12 ccc EEE

xy

z

Ec2 Ec1Inside:

e

cc mk

EkE2

22

11

Outside:

e

cc mk

EkE2

22

22

Inside:

rErm

ErEriEe

cc

11

22

1111 2ˆ

Outside:

rErm

ErEriEe

cc

22

22

2222 2ˆ

Inside:

zki zeyxfAr ,11

Outside:

zki zeyxfAr ,22

Assumed solutions:

1cE

2cE

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

xy

z

xy

z

Ec2 Ec1

Semiconductor Quantum Wires

yxfmk

EEyxfymxm

yxfEyxfymxmm

kE

eyxfEeyxfm

E

e

zc

ee

eee

zc

zikzik

ec

zz

,2

,22

,,222

,,2

1

22

112

22

2

22

112

22

2

2222

1

11

22

1

Inside:

Plug in the assumed solution:

Outside:

yxfmk

EEyxfymxm e

zc

ee,

2,

22 2

22

222

22

2

22

Boundary conditions at the inside-outside boundary:

boundary2boundary1 ,, yxfyxf

boundary2boundary

1 ˆ.,1

ˆ.,1

nyxfm

nyxfm ee

is the unit vector normal to the boundaryn̂

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ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Solve these with the boundary conditions to get for the energy of the confined states:

........3,2,12

,22

1 pmk

EEkpEe

zpczc

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

xy

z

xy

z

Ec2 Ec1

Semiconductor Quantum Wires

zk

1cE

11 EEc

21 EEc 31 EEc

E

The 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 gas

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:

We can add the electron present in each subband as follows:

pfzc

z EkpEfdk

n ,2

2

fE

If we want to write the above as:

fQWE

EEfEgdEnc

1

Then the question is what is the density of states gQW(E ) ?

zk

1cE

11 EEc

21 EEc 31 EEc

E

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ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Start from:

e

zpczc m

kEEkpE

2,

22

1

And convert the k-space integral to energy space:

fp

pcpc

e

E

pf

pc

e

EE

EEfEEEEEE

mdE

EEfEEE

mdEn

c

pc

11

22

122

2

2

1

1

This implies:

EgQW

1cE 11 EEc 21 EEc 31 EEc

Semiconductor Quantum Wires: Density of States

fE

zk

1cE

11 EEc

21 EEc 31 EEc

E

pfzc

z EkpEfdk

n ,2

2

pc

p pc

eQW

EEE

EEE

mEg

1

122

2

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Semiconductor Quantum Wire Lasers

GaAs/AlGaAs quantum wires grown by electron waveguide confinement

metal

A Ridge Waveguide Laser Structure

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ECE 407 – Spring 2009 – Farhan Rana – Cornell University

0D Nanostructures: Semiconductor Quantum Dots

TEM of a PbS quantum dot

Core-shell colloidal quantum dots (Mostly II-VI semiconductors)

CdTe

CdSe

GaAs substrate

InAs quantum dots (MBE)

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Semiconductor Quantum Dots

12 ccc EEE

Ec2 Inside:

e

cc mk

EkE2

22

11

Outside:

e

cc mk

EkE2

22

22

Inside:

rErm

ErEriEe

cc

11

22

1111 2ˆ

Outside:

rErm

ErEriEe

cc

22

22

2222 2ˆ

Inside: zyxfAr ,,11

Outside:

zyxfAr ,,22

Assumed solutions:

Ec1

1cE

2cE

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ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Boundary conditions at the inside-outside boundary:

boundary2boundary1 ,,,, zyxfzyxf

boundary2boundary

1 ˆ.,,1

ˆ.,,1

nzyxfm

nzyxfm ee

is the unit vector normal to the boundaryn̂

Semiconductor Quantum Dots

Ec2

Ec1

Solve these with the boundary conditions to get for the energy of the confined states:

........3,2,11 pEEpE pcc

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

In the limit Ec ∞ the lowest energy level value for a spherical dot of radius R is:

22

1 2

RmE

e

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:

We can add the electron present in each level as follows:

p

fc EpEfN 2

If we want to write the above as:

fQDE

EEfEgdENc

1

Then the question is what is the density of states gQW(E ) ?

Ec2

Ec1

p

cQD pEEEg 2

Because the dot is such a small system, at many times concept of a Fermi level may not even be appropriate!!

EgQD

1cE 11 EEc 21 EEc 31 EEc

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ECE 407 – Spring 2009 – Farhan Rana – Cornell University

A ridge waveguide quantum dot laser structure

metal

Semiconductor Quantum Dot Lasers (III-V Materials)

non-radiativerecombination

electrons

N-doped

photon

stimulated andspontaneousemission

holes

P-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 in direct current modulation• Ability to control emission wavelength via quantum size effect

InAsQdots

InGaAs

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Colloidal Quantum Dots: Wonders of Quantum Size Effect

Photoluminescence from CdSe/ZnS (core-shell colloidal) quantum dots of different sizes (~2-6 nm) pumped with the same laser

Photoluminescence from CdTe/CdSe (core-shell colloidal) quantum dots of different sizes

CdSe

CdTe

1cE

1vE

E

EgQD

photon

photon

RmEE

RmEE

hvv

ecc

21

21

2

22

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ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Quantum Dots: Biology Applications

Motion of quantum-dot-attached-RNA into cells monitored by the luminescence (the quantum dots used are CdSe (core) and ZnS (shell)

CdSe/ZnS quqntum dot coated with DHLA and functionalized with maltose binding protein (MBP) and Avidin

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Quantum Dots: Biology Applications

Nat. Biotechnol., 22, 198-203 (2004)

Invitro microscopy of the binding of EGF to erbB1

erB1 bound to eGFP (enhanced green fluorescent protein)EGF (epidermal growth factor) bound to quantum dotMovie shows binding of EGF tagged with fluorescent quantum dots to erB1 tagged with the green fluorescent protein

CdSe

ZnS

Polymer coating

Nat. Biotechnol., 22, 969 (2004)

Imaging of antibody (PSMA) coated quantum dots targeting cancer tumors cells

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ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Colloidal Quantum Dot Electrically Pumped LEDs

-1

-2

-3

-4

-5

-6

-7

-8

NiO

WO

3

Zn

Cd

S

Cd

Se

Zn

Cd

Se

Sn

O2

(ZT

O)

Zn

Sn

O2

Zn

O

Zn

S

0

Possibleelectron injection layers

Possiblehole injection layers

Quantum dot(QD)

eVVacuum level

Bulovic et. al. (2010)