Quantum Transport:Quantum Transport:Atom to Transistor

Prof. Supriyo DattaECE 659Purdue University

Network for Computational Nanotechnology

02.14.2003

Lecture 14: Basis Functions, Density Matrix IRef. Chapter 4.3

• Summary of Equations:

and

• Recall the concept of a Hilbert Space, a function space conceptually similar to a vector space. Basis functions in Hilbert space are like the unit vectors in vector space.

• Proper use of basis functions facilitate matrix reduction and so are very useful computationally.

• Computationally we often use a non -orthogonal basis set (for example recall H 2). Conceptually this can cause many problems The usual procedure is to transform the oblique sets to orthogonal sets. • From now on it will be assumed that we are working with orthogonal basis sets:

( ) ( )∑=Φ

Φ=Φ

mmm

op

rur

HEvv φ

( ) ( )∑ ′′=Φi

ii rur vv φ

( )∑=′m

mmii ruCu v

{ } [ ]{ }[ ] ACCA

C+=′

Φ′=Φ

zyx ˆ,ˆ,ˆ

mnmnuurd δ=∫ *v

Retouch on Concepts00:00

• Basis transformations are defined by the relation:

Where:

So to transform a vector to a new basis set we’ll have:

• In matrix notation, basis transformations are defined by:

• To show this, we have:

But

Thus, we see that C can be used to transform matrix operators as well as vectors.

( ) ( )∑=Φm

mm rur vv φ ( )∑ ′′=i

ii ru vφ

( )∑=′m

mmii ruCu v

{ } [ ]{ }Φ′=Φ C

[ ] [ ] [ ]CACA +=′

)()(* ruArurdA jopiijvvv ′′=′ ∫

opmmim

AruCrd )(** vv ∑= ∫ )(ruC nnjn

v∑

njmnminm

CAC*∑∑=

( )immi CC +=*

( ) [ ]ijnjmnimnm

ij ACCCACA ++ =∑∑=′∴

Basis Transformations04:08

• First, recall the definition of a unitary transformation:

- Vector length is preserved- Proviso on unitary matrix: C +C= CC+ = I

• Consider the process of finding the eigenvalues and eigenvectors of a Hamiltonian matrix.

• In matlab, we invoke:[V,D]= eig (H)

where D is a diagonal matrix with the eigenvalues on the diagonal and V is a square matrix with the eigenvectors as its columns.

• One way to visualize this process is to consider [H] [D]as a basis transformation from the real space basis to the eigenvector basis. Formally this is expressed as:

[D]= V+ [H] [V]

V=

1 2 n

Hamiltonian Matrix Transformation

11:10

• How do we know it is D= V+HV and not D=VHV+?• Look at the organization of old and new basis sets in V, H and D…

• Rules of matrix multiplication require that the columns of one matrix match the rows of next matrix.• So by observation it must be D= V+HV

V = H = D =

new basis

old basis

old basis

old basis

new basis

new basis

Quantum Transport – Prof. Datta HD

x x=new basis

new basis

old basis

old basis

new basis

old basis

old basis

new basis

Ordering of the Hamiltonian Matrix

18:16

• As an example, we will calculate the electron density of the ‘electrons in a box’

• Remember, by the method of finite differences (using box boundary conditions).

+2t0 -t0H = -t0 +2t0

• We want to find the electron density

Note: ‘occ. a’ refers to the sum over all occupied states. It is very important to not that here the states are either full or empty.

( ) ∑=α

αφ.

2)(occ

xxn

OOOElectrons in a box

x

H

E

x

(ev)

eigenvalue #

…1 2 3 100

100Å Box with

100 pt. lattice

x

Electrons in a Box22:38

• We can redefine n(x) by applying the Fermi function to all states. Where the Fermi function provides the “degree of occupation”between 0 and 1 of a given state at a known electrochemical potential µ.

• So the electron density is:

• We can also re-write this as:

whereif

if

• Note: forms a diagonal matrix called the density matrix.

Occupied and unoccupied levels at µ

E(ev)

eigenvalue #

µ

unoccupied

occupied

( ) ∑=α

ααφ fxxn 2)(

)()( * xfx αα

αα φφ∑=

( ) )()( * xxxn βαβαβα

φρφ∑∑=

αβρ

=αβραf βα =

βα ≠0

Electron Density25:48

• Generalizing we write:

Where n(x) is the diagonal of

• This relation can be seen to represent a unitary transformation from the eigenvector basis to realspace. Note: are given by columns of V:

• In other words:

• Summarizing,

- is in realspace

- is in the eigenstate space

-The diagonal elements of are equal to the electron density n(x).

( ) )()(,~ * xxxx ′∑∑=′ βαβαβα

φρφρ

( )xx ′,~ρ

( )xαΦ

a

( ) xx =αφ α,xV=… …

( ) ( ) *,,

*βββ xx VVx ′′

+ ==′Φ

ρ~VV

VV

ρρ

ρρ~

~

+

+

=

=

ρρ~

( ) ( ) xVx ,αα =Φ

Density Matrix31:00

• So, in the eigenstate basis or “space” ? is a diagonal matrix with elements

• Now in general let us denote the density matrix for any space as ?, where ? is given by:

• What is meant by ?

More Generally, how is the ‘function’ of a matrix calculated? For a diagonal matrix it is simply the ‘function’ operated on all elements. How about matrices with off diagonal elements?

• Example: Given [H] with off diagonal elements calculate (sin[H]). To do this we must first diagonalize [H], then operate sin () upon the diagonalized form of , and then finally transform [H] back into its original space.

TkE Bef /)(1

1µα α −+

=

( )µα −= Ef0

[ ] [ ]( )IHf µρ −= 0

[ ] [ ]( )IHf µ−0

[ ]H

Equilibrium Density Matrix36:30

Example: continued.

(1) Diagonalize [H]

H1H2

(2) Operate sin ()

sin(H1)sin(H2)

(3) Transform back to original space

sin(H1)V sin(H2) V+

• Note: In matlab matrix functions and element by element functions are differentiated by addition of an ‘m’ for matrix functions. e.g. :sin () : represents element by element operation sinm (): represents matrix operations.

• Finally we see the expression for ? in real space is:

• Interestingly, ? is only diagonal in the eigenvector basis. Off diagonal elements of ? in alternate basis sets are used in some calculations, but more often than not only the diagonal elements of ? (which in any space provide electron density) are of interest.

O

O

O

V=ρ~( )µ−10 Ef

( )µ−20 Ef

( )µ−nEf0

O+V

Equilibrium Density Matrix42:00