Quantum Transport :Atom to Transistor,What Makes Electrons Flow

8/8/2019 Quantum Transport :Atom to Transistor,What Makes Electrons Flow

1/11

Prof. Supriyo DattaECE 659Purdue University

Network for Computational Nanotechnology

01.15.2003

Quantum Transport:Quantum Transport:Atom to Transistor

Lecture 2: What Makes Electrons Flow?

Ref. Chapter 1.2


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01.15.2003

00:08

Key Concepts: VD and VG Bias, Empty andfull energy levels, Threshold voltage VT, andFermi energy

Basic Fermi function centered at E=0: f0(E)

= 1/ (e E / kBT + 1). To get it centered at Ef, one

could shift it in energy: f0(E-EF)

0eV

EF

EC

Vacuum Level

EV

Empty

fo0 1

E

ID

VGVT

DRAIN

SOURCE

I

VG VDGate

INSULATOR

CHANNEL

L

INSULATORz

x

Retouch on Concepts


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05:05

At equilibrium, there is no bias voltage

between the source and drain, the Fermi

energy is fixed at the same level in source,channel and drain

Very Important: The amount of currentflow does not depend on number of

electrons. It is the amount of states around

the Fermi energy that determines how

much current flows.

0eV

EF

EC

Vacuum Level

EV

Empty

f00 1

E

Pauli Exclusion Principle allows

two electrons per energy level

(one spin up and one spin down)

Positive gate voltage lowers

these levels and negative gate

voltage raises them

D

RAIN

S

OURCE

IVG VDGate

INSULATOR

CHANNEL

L

INSULATOR

zx

Retouch on Concepts


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01.15.2003

Consider a simple system with wide spacing of electron energy levels (a small

molecule)

4 electrons in the system considering spin up and down fill the first 2 energy

levels. The Fermi energy is a level for which at T=0K the levels below it are full and the

levels above it are empty. For our system at 0 K, the Fermi level will be half way

between levels 2 and 3.

09:22

EF

E4E3

E2E1electrons

The smaller the system, the more the

spacing between the energy levels.

The picks observed in the current are due to the

overlap of energy levels and the Fermi level: As the gate

voltage is applied these energy levels move up or down

based on the sign of the voltage; hence they cross over the

Fermi level and the result will be higher current because ofhigher availability of states at the Fermi energy.

Band Diagram for aSimple Molecule

ID

VG

Fixed Small

VD


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12:32

When VD is applied between the source and

drain, system is driven out of equilibrium; theFermi energies in the source and drain will

not be at the same level any more.

Total energy difference between the two

Fermi levels (chemical potentials) 1 and 2 is

qVD. i.e. 1Vx q= 1eV= 1.6 x 10-19J

The reason why the shift in

Fermi levels is symmetric will

be discussed later. For now

consider the fact that division

of voltage is very important for

large bias but not so important

for small bias which is in factthe case here.

qV

Vacuum Level

1=Ef+ qVD/2

2=Ef- qVD/2

DRAIN

SOURCE

IVG VDGate

INSULATOR

CHANNEL

L

INSULATORz

x

D

Away from Equilibrium


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16:59

Why dont bottom and top levels contributeto current? Since we have two different Fermi

levels, we have two different Fermi functions.

Consider their value for high energies and for

low energies; youll find that they have the

same value of 0 (For high E) and 1 (For low

E) - In other words, both contacts have the

same agenda for these levels; hence nocurrent flow due to these levels.

E

2/1 DF qVE +=

Vacuum Level0eV Top Levels

Bottom Levels

2/2 DF qVE =E

If you do not have any

levels between 1 and 2

then current will not flow

If you have a level between

1 and 2 current will flow

because left side (at )

keeps filling up the level and

right side (at ) keepsemptying it.

1

2

Current Flow due toDifference in Agenda


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24:19

What is the average number of

electrons in the level ?

N1: the average number ofelectrons that the leftcontact would

like to see

N1= 2f1 () = 2f0 ( - 1)

N2: the average number of

electrons that the rightcontact would

like to see

N2= 2f2 (

) = 2f0 (

- 2)

Note: factor of two is due to the fact thatwe can put two electrons of up and down

spin in the same level.

Vacuum Level

1

2

0eV

I1

I2

Average Number ofElectrons


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28:28

N is the actual number of electrons at steady state in

the channel

Current from left side:

Where is the rate at which electrons cross

from 1 to . Current to right side:

Where is the rate at which electrons cross

from to 2 .

So are in units of Joules

Take , we have:

Vacuum Level

1

2

0eV

I1

I2

)( 11

1 NNqI =h

h1

)( 22

2 NNqI =

h

h2

sec1006.1

2

34 == Jh

h

21 +

meV11 =

sec10sec1006.110106.1

12

34

319

1

=

JJh

Therefore, it takes

~1 pico second for

electrons to escapeinto the channel

Current Flow Left andRight


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33:55

Equating I1 and I2 for a steady state solution we

get

Remember, in order for current to flow, f1 mustdiffer from f2 N-Type conduction: Go through a level that is

empty at equilibrium

P-Type conduction: Go through a level that is

full at equilibrium

Vacuum Level

1

2

0eV

I1

I221

2211

+

+=

NNN

)( 2121

2121 NN

qIII ===

+

h

)]()([2

21

21

21 ffq

=+

h

Steady State Current


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01.15.2003

38:37

Use Taylor expansion to get an expression for current at

small voltages: f1() = f0( - 1)

f2

() = f0

( - 2

) therefore:

f1- f2 = (df0 / dE) (2 - 1)

= -(df0 / dE) qVD

Therefore for small voltages:

Use E = -Efsince 1= Ef+ qVD/2 and 2= Ef qVD /2

Vacuum Level

1

2

0eV

I1

I2

fEEdE

dfqV

=+

0

21

21

22

h

Note: , dimensions of conductance

, dimensions of energy

, dimensions of inverse energy

h

22q

21

21

+

fEEdEdf

=

0

== + )]()([2

21

21

21 ffq

I

h

Current at Small Voltages


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01.15.2003

Peak 1/(4kBT) ; Width kBT Area

= 1.

As t0, it can be approximated as a

delta function.

What does look like?

46:01

Ef

f01

E E

dE

df0

1/(4kBT)

dE

df 0 From the equation

it would appear that there is no upper

limit on the conductance

However, this is not true. An upper

limit does occur due to energybroadening. The maximum value ofconductance for one level device is:

to be discussed in more detail next

fEEdE

dfqVI

=+

=

0

21

21

22

h

fEEdEdfq

=+

0

21

21

2

2h

kS

h

q

9.12

14.772 2

No Upper Limit?

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Quantum Transport :Atom to Transistor,What Makes Electrons Flow