Modelowanie Nanostruktur

1!

Modelowanie Nanostruktur

Semester Zimowy 2012/2013 Wyk!ad

Jacek A. Majewski Chair of Condensed Matter Physics Institute of Theoretical Physics

Faculty of Physics, Universityof Warsaw

E-mail: Jacek.Majewski@fuw.edu.pl

Modelowanie Nanostruktur, 2012/2013 Jacek A. Majewski

Wyk!ad 3 – 9 X 2012

Metody ci"g!e w modelowaniu nanostruktur

Metoda masy efektywnej dla studni kwantowej

Nanotechnology –

Low Dimensional Structures

Quantum

Wells Quantum

Wires Quantum Dots A B

Simple heterostructure

What about realistic nanostructures ?

2D (quantum wells): 10-100 atoms in the unit cell 1D (quantum wires): 1 K-10 K atoms in the unit cell 0D (quantum dots): 100K-1000 K atoms in the unit cell

Organics

Nanotubes, DNA: 100-1000 atoms (or more)

Inorganics

3D (bulks) : 1-10 atoms in the unit cell

2!

Synthesis of colloidal nanocrystals Injection of organometallic precursors

Mixture of surfactants Heating mantle

Nanostructures: colloidal crystals!

- !Crystal from sub-!m spheres of PMMA (perpex) suspended in organic solvent;

-! self-assembly when spheres density high enough;

0 2 4 6 8 10 12 14 1

10 100 1 000 10 000 100 000 1e+06

Number of atoms

R (nm)

Tight-Binding

potential Ab initio

Atomistic vs. Continuous Methods

Microscopic approaches can be applied

to calculate properties of realistic nanostructures

Number of atoms in a spherical Si nanocrystal as a function of its radius R.

Current limits of the main techniques for calculating electronic structure.

Nanostructures commonly studied experimentally lie in the size range 2-15 nm.

Continuous methods

Atomistic methods for modeling of nanostructures

Ab initio methods (up to few hundred atoms)

Semiempirical methods (up to 1M atoms)

(Empirical Pseudopotential) Tight-Binding Methods

(e.g., effective mass approximation)

3!

Continuum theory- Envelope Function Theory

Electron in an external field

!ˆp

2m + +V (!r) ++U(!r)

!!

""

## ##

$$

%%

&&

&& ! ! (!r) == !!" " (!r)

Periodic potential of crystal Non-periodic external potential

Strongly varying on atomic scale Slowly varying on atomic scale

0

-5 5

!1

"3

"1

"1

"1

"₃

# ‘2

# ‘25

#15

#1

$ ‘2

$ ‘2

$₅

$1

$5

L ‘2 L1 L ‘3 L3

L1 !4

!1

Energy [eV]

Wave vector k

$1

"₁

Ge

Band structure of Germanium

!!

( ! k) U( _{r ) == 0} !

Electron in an external field

!ˆp

2m + +V (!r) ++U(!r)

!!

""

## ##

$$

%%

&&

&& ! ! (!r) == !!" " (!r)

Periodic potential of crystal Non-periodic external potential Strongly varying on atomic scale Slowly varying on atomic scale

Which external fields ?

!! Shallow impurities, e.g., donors

!! Magnetic field B,

!!Heterostructures, Quantum Wells, Quantum wires, Q. Dots U(!r) == !! e²

!!| !r | B == curl! !

A ==!

!

! ""! A

GaAs GaAlAs cbb

GaAs

GaAlAs GaAlAs

Does equation that involves the effective mass and a slowly varying function exist ? !ˆp²

2m*++U(!r)

!!

""

#### $$

%%&&&&F(!r) ==!!F(!r)

F(!r) == ?

Envelope Function Theory – Effective Mass Equation

J. M. Luttinger & W. Kohn, Phys. Rev. B 97, 869 (1955).

[ !! (!!i !

"

") ++U(!r) !! !! ]F

(!r) == 0

!

! (!r) == F

(!r)u

(!r)

U(!r) == 0

(!r) == exp(i !

(EME) EME does not couple different bands

Envelope Function

Periodic Bloch Function

“True”

wavefunction

Special case of constant (or zero) external potential

!

! (!r)

( )

U z

(!r) == exp[i(k

(z)

4!

Electronic states in Quantum Wells

Envelope Function Theory- Electrons in Quantum Wells

Let us consider an electron in the conduction band near point

! !

GaAs

GaAlAs GaAlAs

cbb

Growth direction (z – direction )

Potential ? U(!r)

U(!r) is constant in the xy plane

!!_c(!

k) ==!!_c0++ !² 2m^*

k!²

U(!r) == U(z) == !!_c0 z !! GaAs

!!_c0++!!E_c z !! GaAlAs

""

##$$

%%$$

!! !² 2m*

!!²

!!x²++!!²

!!y²++!!²

!!z²

""

##$$$$ %%

&&

''''F(x, y,z)++U(z)F(x, y,z) == EF(x, y,z) Effective Mass Equation for the Envelope Function F

( , , ) _x( ) ( ) ( )_{y z} F x y z ==F x F y F z Separation Ansatz

!! !² 2m*

""²F_x

""x² F_yF_z++""²F_y

""y² F_xF_z++""²F_z

""z² F_xF_y

##

$$

%%%%

&&

''

((((++U(z)F_xF_yF_z==EF_xF_yF_z

!! !² 2m*

""²F_x

""x² F_yF_z++""²F_y

""y² F_xF_z++""²F_z

""z² F_xF_y

##

$$

%%%%

&&

''

((((++U(z)F_xF_yF_z==EF_xF_yF_z

x y z

E E== ++E ++E

!! !² 2m*

!!²F_x

!!x² F_yF_z==E_xF_xF_yF_z !! !² 2m*

!!²F_y

!!y² F_xF_z==E_yF_xF_yF_z

!! !² 2m*

""²F_z

""z² F_xF_y++U(z)F_xF_yF_z==E_zF_xF_yF_z

!! !² 2m*

!!²F_x

!!x² ==E_xF_x !!F_x~ e^ikxx, E_x== !² 2m*k_x²

Effective Mass Equation of an Electron in a Quantum Well

!! !² 2m*

!!²F_y

!!y² ==E_yF_y !!F_y~ e^ikyy, E_y== !² 2m*k_y²

!! !² 2m*

!!²F_zn

!!z² ++U(z)F_zn==E_znF_zn

Conduction band states of a Quantum Well

||(!r_||) == 1

Aexp[i(k_xx ++ k_yy)] == 1 Aexp(i!

k_||!!!r_||) F_n,k^!

||(!r) == F_k^!

||(!r_||)F_zn(z) == 1 Aexp(i!

k_||!!!r_||)F_zn(z) E_n(!

k_||) == !² 2m*

"

k_||²++E_zn Energies of bound states in Quantum Well

!! !² 2m*

!!²F_zn

!!z² ++U(z)F_zn==E_znF_zn

Energy

n=1 n=2 n=3

0

!!

c0 Ec

!! ++""

Ezn

E

( !

k

)

k !

E == E₁++ !² 2m*

k!_||² E == E₂++ !²

2m*

k!_||²

E

E

Wave functions F_zn(z) Ezn

zn( ) F z

5!

Band structure Engineering

The major goal of the fabrication of heterostructures is the controllable modification of the energy bands of carriers

Lattice constant

Energy Gap [eV]

Band lineups in semiconductors!

-12

-10

-8

-6

-4

0

SiC

Si

Ge

InSb

GaSb

InAs

AlSb

GaAs

AlAs

InP

GaP

AlP

InN

GaN

AlN

ZnO

ZnS

CdS

ZnSe

CdSe

ENERGY (eV)

HgTe

CdTe

ZnTe

Band edges compile by: Van de Walle (UCSB) Neugebauer (Padeborn), Nature’04

e.g., GaAs/GaAlAs InAs/GaSb

Band lineup in GaAs / GaAlAs Quantum Well with Al mole fraction equal 20%

Envelope Function Theory- Electrons in Quantum Structures

Type-I Type-II

cbb

A B E

!

!

Ec

!

!

gapA E

gapB

E vbt

cbb A B

E

!

!

Ec

!

!

gapA

E

gapB

E vbt

cbb A

v

B E

!

!

Ec

!

!

gapA

E E^gapB

Various possible band-edge lineups in heterostructures

GaN/SiC

(staggered)

(

)

- Valence Band Offset (VBO)

E

!

!

GaAlAs vbt

cbb

1.75 eV 1.52 eV 0.14 eV

0.09 eV GaAs

VBO’s can be only obtained either from experiment or ab-initio calculatio

Effective Mass Theory with Position Dependent Electron Effective Mass

* *

A B

m !!m

*A

m m^*_B 0 z ==

!! !² 2m*(z)d²

dz² !!!²

2 d dz 1

m*(z)d dz

!!!² 2 d dz

1 m*(z)

d dz

!!

""

## $$

%%&&F(z) ++U(z)F(z) == EF(z)

ˆ ˆ

[ *( )] _z[ *( )] _z[ *( )]

T== m z ^!!p m z ^""p m z ^!! 2!! ""++ == ##1 ( )

F z _{m z* ( )}^{1 dF z}_dz^{( )} z ==0

*A

m

*( ) m zB

“Graded structures”

IS NOT HERMITIAN !!

Symetrization of the kinetic energy operator

General form of the kinetic energy operator with

IS HERMITIAN !

and ARE CONTINOUOS !

6!

Doping in Semiconductor Low Dimensional Structures

E

!

! E

!

!

Energy band diagram of a selectively doped AlGAAs/GaAs Heterostructure before (left) and after (right) charge transfer

AlGaAs

GaAs VACUUM LEVEL

Ev

!

!

E_F

gA

E

Bg

E

!!A

!!B

gA

E

!!

LA

Negatively charged region Positively charged region

!!Aand - The electron affinities of material !!_B A & B

The Fermi level in the GaAlAs material is supposed to be pinned on the donor level.

The narrow bandgap material GaAs is slightly p doped.

gB

E

( ) (0)( ) ( ) U z ==U z e z++ !!

Effects of Doping on Electron States in Heterostructures +

+

E_c

+ +

+

+ + + + +

Unstable Charge transfer Thermal equilibrium

!

!²!!(!r) == 4!!e

!! ""(!r "" !R_A) "" !!(!r "" !R_D) "" f_!!|""_!!(!r) |²

!!

#

(don)

# # #

(acc)

# #

$$

%%

&&

&&

'' ((

)) )) Resulting electrostatic potential

should be taken into account in the Effective Mass Equation

!! !²

2m*

""²

""x²++""²

""y²++""²

""z²

##

$$%%%% &&

''((((++U(x, y,z)!! e!!(!r)

))

**

++++

,, -- ..

..!!_""(!r) == E_!!""_!!(!r)

!

!²!!(!r) == 4!!e

!! ^NA(!r) "" N_D(!r) "" f_!!|""_!!(!r) |²

!!

#

$$

#

%%

&&

&&

'' ((

))))

Electrostatic potential can be obtained from the averaged acceptor and donor concentrations

Fermi distribution function

The self-consistent problem, so-called “Schrödinger-Poisson” problem

To be continued!