Upload
liakman
View
224
Download
0
Embed Size (px)
Citation preview
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
8/8/2019 Quantum Transport :Atom to Transistor,What Makes Electrons Flow
2/11
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
8/8/2019 Quantum Transport :Atom to Transistor,What Makes Electrons Flow
3/11
01.15.2003
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
8/8/2019 Quantum Transport :Atom to Transistor,What Makes Electrons Flow
4/11
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
8/8/2019 Quantum Transport :Atom to Transistor,What Makes Electrons Flow
5/11
01.15.2003
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
8/8/2019 Quantum Transport :Atom to Transistor,What Makes Electrons Flow
6/11
01.15.2003
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
8/8/2019 Quantum Transport :Atom to Transistor,What Makes Electrons Flow
7/11
01.15.2003
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
8/8/2019 Quantum Transport :Atom to Transistor,What Makes Electrons Flow
8/11
01.15.2003
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
8/8/2019 Quantum Transport :Atom to Transistor,What Makes Electrons Flow
9/11
01.15.2003
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
8/8/2019 Quantum Transport :Atom to Transistor,What Makes Electrons Flow
10/11
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
8/8/2019 Quantum Transport :Atom to Transistor,What Makes Electrons Flow
11/11
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?