Quantum Transport :Atom to Transistor,Supplementary Topic: Spin

8/8/2019 Quantum Transport :Atom to Transistor,Supplementary Topic: Spin

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Quantum Transport:Quantum Transport:Atom to Transistor

Prof. Supriyo DattaECE 659Purdue University

Network for Computational Nanotechnology

04.30.2003

Lecture 42: Supplementary Topic: SpinRef. Chapter 5.4 & 5.5


2/21

In this lecture, well discuss

electron spin

Recall, when deriving current

through a small structure, like

that show on the right, we first

obtain a Hamiltonian [H].

Usually, however, theeigenenergies of [H] represent

two degenerate spin levels

The proper Hamiltonian for degenerate

spin levels is twice as big

with no coupling between the spin up andspin down portions

Small Device

1 2

[H]

HO

OH

Spin00:00


3/21

We want to consider the conditions

for which the spin levels are not

degenerate. There are two cases

by which this arises

Case 1: Strong Electron-Electron

Interactions (i.e. Coulomb Blockade)

Because electrons are forced to

interact strongly instead of 2

degenerate levels

we have two split levels

the first level fills

and forces the

second up

Important Example: Magnetism.

All electrons want to have the same

spin due to electron-electroninteractions.

Strong ElectronInteractions

02:20


4/21

Case 2: Spin Orbit Interaction

This has nothing to do with

electron-electron interactions and isthe focus of the remainder of this

lecture.

Continuing with spin-orbit

interaction

Ordinarily the Hamiltonian is

When a magnetic field is applied,

say in the direction, up and down

spin levels do not remain degenerate

so we add

where B is the Bohr magneton

constant . A well knownexample of this is the Zeeman effect

+

+

UmpO

OUmp

2

22

2

Z

Z

B

B

B

0

0

z

mq 2/h

Spin Orbit Interaction06:08


5/21

If we apply a magnetic field in the x-

direction as well the additional term

becomes

To summarize, the magnetic field

gives

and the magnetic field gives

These matrices have the same

eigenenergies but different

eigenvectors. The eigenenergies

are

+BBz, -BBz,

+BBx, -BBx

x

01

10xBB

+

ZX

XZ

BBB

BB

z

1001

ZBB

Eigenenergies08:55


6/21

The eigenvectors are

direction

direction

One might imagine that we could

have written the additional term as

this model does explain theZeeman effect but fails under what

is known as the Stern-Gerlach

experiment

+

+22

22

0

0

xz

xz

B

BB

BB

x

z +BBz

-BBz

0

1

1

0

+BBx

-BBx

1

1

1

1

Note the different eigenvectors

Eigenvectors11:15


7/21

Example Continued:

And if we add another

inhomogeneous, say in the x-

direction, the beam splits again

In the Stern-Gerlach experiment a

beam of electrons is injected into an

inhomogeneous magnetic field, due to

spin interaction the beam splits after

entering the magnetic field.

Example: A beam of electrons split by

an inhomogeneous magnetic fieldz

0

1

1

0

Bz

high

low

1

1Bz

high

low

1

1

1

1

1

1

Bx

high

low

Stern-Gerlach Experiment12:39


8/21

Overall the correct term,

valid for any direction, added

to the Hamiltonian is

Note, regardless of magnetic

field direction the splitting

effect is the same though theeigenvalues may vary

Often this matrix is written as

where

and

+

ZyX

yXZ

BBiBB

iBBB

Bvv

zyx

zyx

BzByBxBzyx

++=++=v

v

+=

=

=

=

zyx

yxz

yxz

BiBBiBBBB

i

i

vv

0

0

01

10

10

01

Eigenenergies16:59


9/21

Moving on we can define

spin-orbit interaction

Recall, an electric field may

be expressed as

and a magnetic field as

A scalar potential, U, is

simply added into the

Schrdinger equation

But a vector potential, , is incorporated

as a dot product with the momentum vector

, i.e.

t

AE

=

vv

AB

vvv

=

Av

pv

( )

( )

++

++

Um

Aqp

U

m

Aqp

20

0

2 2

2

vv

vv

Vector Potential19:40


10/21

Interestingly, even when there is

nomagnetic field a strong electric

field can induce an effectivemagnetic field on an electron. This

phenomena occurs due to

relativistic interactions between an

electron and the applied electric

field. Interaction between the

electric field of the nucleus and

electrons of an atom is a very

good example of this. We call this

effect Spin-Orbit Interaction

Spin-Orbit Interaction is incorporated

into the system Hamiltonian with the term

Note: Even in a weak electric field spin-

orbit interaction is present but generally

is too small to deserve any consideration

22mc

pEB

vvv

Spin-Orbit Interaction23:42


11/21

Spin-Orbit coupling plays a

fundamental role in our understanding

of semiconductor physics

Given a semiconductor with an E-k

diagram like the following

If we were to examine the area

right around the dot we would find

that conduction band was formed of

|s> orbitals and the valence band of

|p> orbitals (|px>,|py>,|pz> or in

another notation |x>, |y>, |z>)E

k

Semiconductors25:34


12/21

Ignoring spin orbit coupling one might

assume semiconductor valence bands

of the form |x>, |y>, and |z> separately.If this were the case then a transition

from the conduction band, |s>, to say

|x> would be polarized in the x

direction. Furthermore, since there is

no preference in transition to the

valence band y-polarized and z-

polarized light should be equally likely

resulting in overall isotropic emission

However, this is not the case,

emission is not isotropic but in fact

circularly polarized. We cannotignore spin-orbit

coupling/interactions. When

included, spin-orbit coupling

provides an entirely different (but

correct) set of valence bands that

accurately predict the circularly

polarized light emitted from in

semiconductors

Isotropic Emission27:24


13/21

The first valence band

derived with spin-orbit

coupling is

and

we call this the heavy hole

band. We will ignore the

other valence bands (light

hole) for now since most

optical transitions occur to

and from the heavy hole band

Heavy Hole Band

E

k

Note: The |s>

conduction orbital is

not usually affected by

spin-orbit coupling

+

0

yix

yix

0 Ex:

will produce circularlypolarized light in the x-

y plane. So we see

what a dramatic

physical effect spin-

orbit coupling can have

+

00

yixsto

Correct Valence Bands29:03


14/21

Important Point: Even when

electrons travel at a fraction of

the speed of light relativistic

phenomena can produceobservable effects on spin

Note: In somesemiconductors, such as InAs,

spin-orbit interaction can alter

even the conduction band

So how does one make sense of the spin-

orbit interaction term

To begin, look at the vector term in theHamiltonian (ignoring the spin-orbit term

and any scalar potential U):

where I is a 2x2 matrix

22mc

pEB

vv

v

BIm

AqpH B

vvvv

+

+=

2

2

Spin-Orbit Interaction34:00


15/21

Note that

is equivalent to

where is composed of the Pauli spinmatrices

Furthermore, without any vector potential

,

( )BI

m

AqpB

vvvv

++

2

2

( )[ ]m

Aqp

2

2vvv+

v

Av

( )

=

=

mp

mp

m

pH

20

02

2 2

22vv

But if we include and apply a

few dot and cross product

identities to the familiar

term in is produced

Point: is related to thegeneral vector Hamiltonian and

does not simply appear from

nowhere

1

2

Av

BB

vv

BBvv

Vector Hamiltonian37:00


16/21

Now, the spin-orbit interaction term

is relativistic and can only be derived from the

full relativistic Hamiltonian

The full relativistic Hamiltonian, otherwiseknown as the Dirac equation, is

Note: I is the 2x2 identitiy matrix, c is the

velocity of light, and is replaced by

when a magnetic field is applied

=

Imcpc

pcImcH

2

2

vv

vv

So how do we get the

familiar, low velocity,

Schrdinger equation

from the Dirac equation?

22mc

p

B

vvv

pv

Aqpvv

+

Im

pH

2

2

=

Deriving the Spin-OrbitTerm

39:51


17/21

Recall, the E-k

relationship at low

electron energies, forwhich

is valid looks like

Whereas the high energy Dirac equation produces

an E-k relationship of the form

Im

pH

2

2

=

E

k

E

k

- - -

-

+--

-when an electron

moves up it leaves

a position

+mc2

-mc2

bottom band

filled with

electrons

E-k Relationships42:00


18/21

E

k

- - - -+--

-when an electronmoves up it leaves

a position

+mc2

-mc2

bottom band

filled withelectrons

Note however, that moving to the top band

requires an energy equal to 2mc2 (on the

order of 1MeV)

For the most part, outside

high energy experiments, only

a few electrons reside in the

top-band. Importantly, we cangreatly simplify the Dirac

equation by concentrating on

the few electrons in the top

band and ignore the bottomband completely. Based on

this assumption we can derive

the familiar low energy

Hamiltonian.

Only a Few at the Top43:44

0


19/21

Conceptually, in the low energy regime we

can concentrate on term in the Dirac

Equation

and treat terms , , and as a self-

energy

Recall, for any system, H, coupled to alarger system, HR, we may define a self-

energy

Coupled Systems

System Large System

H HR

Imcpc

pcImc2

2

vwvw

1 2

3 4

+= 1)( RHE

For the Dirac equation

and

2mcH

pc

R =

== +vv

( ) ( )pcmcEpcvvvv

+= 2

1

Dirac Equation Self-Energy

45:40

47 00


20/21

Furthermore, since the electrons we

are concentrating on, which lie in the top

band, have an energy around mc2 we

may make the approximation E=mc2.Thus,

So, overall the non-relativistic first-

order Hamiltonian is

( ) ( )

( )m

p

pcmc

pc

2

2

1

2

2

vv

vvvv

=

=

( )m

p

ImcH 2

2

2

vv

+=

But mc2I, as constant, may be

ignored which gives the familiar

To include spin-orbit interactions

we need to improve our

approximation for E, that is E

mc2, to something more accurate

( )m

pH

2

2vv =

First Order Self-Energy47:00

48 10


21/21

Including a more accurate representation of E complicates the algebra

considerably, nonetheless by including a next order approximation of E we

are able to derive the familiar spin-orbit interaction term

from the Dirac equation

Main Point: Spin-orbit interactions, magnetic interaction, vector potentials,

etc. all follow from the Dirac equation and are all fundamentally related

22mc

pE

B

vvv

Closing Comments48:10

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Quantum Transport :Atom to Transistor,Supplementary Topic: Spin