ECE 802-604:
Nanoelectronics
Prof. Virginia Ayres
Electrical & Computer Engineering
Michigan State University
ayresv@msu.edu
Lecture 04, 10 Sep 13
In Chapter 01 in Datta:
Two dimensional electron gas (2-DEG)
DEG goes down, mobility goes up
Define mobility
Proportional to momentum relaxation time tm
Count carriers nS available for current – Pr. 1.3 (1-DEG)
How nS influences scattering in unexpected ways – Pr 1.1 (2-DEG)
One dimensional electron gas (1-DEG)
Special Schrödinger eqn (Con E) that accommodates:
Electronic confinement: band bending due to space charge
Useful external B-field
Experimental measure for mobility
VM Ayres, ECE802-604, F13
Lecture 04, 10 Sep 13
In Chapter 01 in Datta:
Two dimensional electron gas (2-DEG)
DEG goes down, mobility goes up
Define mobility m
Proportional to momentum relaxation time tm
Count carriers nS available for current – Pr. 1.3 (1-DEG)
How nS influences scattering in unexpected ways – Pr 1.1 (2-DEG)
One dimensional electron gas (1-DEG)
Special Schrödinger eqn (Con E) that accommodates:
Electronic confinement: band bending due to space charge
Useful external B-field
Experimental measure for mobility
VM Ayres, ECE802-604, F13
Wire up HEMT to use the triangular quantum well region in GaAs
-z
y
Ey
nx
= (-|e |)(-|E y|)
y
z
Correct for e-’s with Drain = +
Note: current I is IDS
VM Ayres, ECE802-604, F13
Why do this: increase in Mobility in using 2-DEG region in GaAs instead
of 3-DEG bulk GaAs
mobility
931C: 3D Scattering
Sweet spot
at 300K
T = cold:
Impurity =
ND+, NAscattering
T = hot:
Phonon
lattice
scattering
VM Ayres, ECE802-604, F13
Increase in Mobility is based on decrease of scattering, or said another
way, increase e-s not scattered.
Scattering involves energy and momentum conserving
interactions. Putting quantum restrictions on these interactions
means that fewer can occur.
VM Ayres, ECE802-604, F13
Streetman t:
Datta tm  t: The statement below is true for a group of e-s not a
single scattering event. tm is an average or mean time
VM Ayres, ECE802-604, F13
Lecture 04, 10 Sep 13
In Chapter 01 in Datta:
Two dimensional electron gas (2-DEG)
DEG goes down, mobility goes up
Define mobility m
Proportional to momentum relaxation time tm
Count carriers nS available for current – Pr. 1.3 (1-DEG)
How nS influences scattering in unexpected ways – Pr 1.1 (2-DEG)
One dimensional electron gas (1-DEG)
Special Schrödinger eqn (Con E) that accommodates:
Electronic confinement: band bending due to space charge
Useful external B-field
Experimental measure for mobility
VM Ayres, ECE802-604, F13
2-DEG: Major improvement in performance at low temperatures
mobility
931C: 3D Scattering
Sweet spot
at 300K
T = cold:
Impurity =
ND+, NAscattering
T = hot:
Phonon
lattice
scattering
VM Ayres, ECE802-604, F13
2-DEG: large increase in carrier concentration nS:
intrinisic
VM Ayres, ECE802-604, F13
2-DEG: large increase in carrier concentration nS:
intrinisic
3-DEG
VM Ayres, ECE802-604, F13
2-DEG: Energy:
Special Schrödinger eqn (Con E) that accommodates:
Electronic confinement: band bending due to space charge
Useful external B-field
Example: ECE874, Pr. 3.5 with E-field: determine direction of motion.
Datta 1.2.1 would be correct way to continue the problem to get energy levels
VM Ayres, ECE802-604, F13
2-DEG: Energy:
2-DEG wavefunction
Use this wave function in the special Schroedinger eq’n and
it will separate into kz and kx, ky parts.
kz is a fixed quantized number(s).
kx, ky are continuous numbers
VM Ayres, ECE802-604, F13
2-DEG: Energy:
For the kx, ky part:
VM Ayres, ECE802-604, F13
Bulk Dimensionality Systems: 3-DEG
z
KE =
px2 + py2 + pz2
2m* 2m*
2m*
y
Free motion in all directions
x
px , py , pz can take any values
Macroscopic World
Bulk Materials
Silicon
Ingot
B. Jacobs, PhD thesis
VM Ayres, ECE802-604, F13
Reduced Dimensionality Systems: 2-DEG
KE
z
E=
px2 + py2 + nz2 ħ2p2
2m* 2m*
2m*Lz2
y
x
Free motion in x and y directions
Shown: Infinite potential well in z direction
Graphene
pz is constrained to be a number(s)
Thin Films
Thin layers
J.S. Moodera, Francis Bitter Magnet Lab, MIT
A.K. Geim and K.S. Novoselov,
Nat. Mater., 2007, 6, 183
B. Jacobs, PhD thesis
VM Ayres, ECE802-604, F13
Reduced Dimensionality Systems: 1-DEG
KE
z
E=
nx2 ħ2p2 + py2 + nz2 ħ2p2
2m*Lx2 2m* 2m*Lz2
y
x
Free motion in y direction
Shown: Infinite potential well in x and z directions
px , pz are constrained to be a number(s)
Carbon Nanotubes,
Nanowires,
Molecular Electronics
1μm
Richard E. Smalley Institute,
Rice University
B. Jacobs, PhD thesis
VM Ayres, ECE802-604, F13
Reduced Dimensionality Systems: 0-DEG
E=
z
x
y
nx2 ħ2p2 + ny2 ħ2p2 + nz2 ħ2p2
2m*Lx2 2m*Ly2
2m*Lz2
No free motion. Enter and leave QD by tunnelling
Shown: Infinite potential well in x, y and z directions
px, py, pz are constrained to be a number(s)
Quantum Dots
A. Kadavanich, MRSCE, University of Wisconsin
B. Jacobs, PhD thesis
VM Ayres, ECE802-604, F13
2-DEG in a semiconductor:
KE
VM Ayres, ECE802-604, F13
2-DEG in a semiconductor:
VM Ayres, ECE802-604, F13
2-DEG in a semiconductor:
You have put integral travelling waves
in a large box but are ignoring the
edges
VM Ayres, ECE802-604, F13
2-DEG in a semiconductor:
Standing waves in a small box.
Edges matter.
VM Ayres, ECE802-604, F13
2-DEG in a semiconductor:
VM Ayres, ECE802-604, F13
2-DEG in a semiconductor:
S
ES is the minimum energy required for an e- to be out of a bond.
VM Ayres, ECE802-604, F13
2-DEG in a semiconductor:
Similar to:
e1
EC = Egap
ES is the minimum energy required for an e- to be out of a bond.
VM Ayres, ECE802-604, F13
2-DEG in a semiconductor:
e1
VM Ayres, ECE802-604, F13
2-DEG in a semiconductor:
e1
kx
Any little patch on there would have some values of kx, ky
VM Ayres, ECE802-604, F13
2-DEG in a semiconductor:
e1
kx
y-axis is E.
The bowl is the KE that an e- has above the minimum requirement of ES
required to be out of a bond
p = hbar k and KE = p2/ 2m
VM Ayres, ECE802-604, F13
Reduced Dimensionality Systems: 2-DEG
KE: write p in terms of hbar k
z
E=
px2 + py2 + nz2 ħ2p2
2m* 2m*
2m*Lz2
y
x
Free motion in x and y directions
Shown: Infinite potential well in z direction
Graphene
pz is constrained to be a number(s)
Thin Films
Thin layers
J.S. Moodera, Francis Bitter Magnet Lab, MIT
A.K. Geim and K.S. Novoselov,
Nat. Mater., 2007, 6, 183
VM Ayres, ECE802-604, F13
Go back to this idea:
You have put integral travelling waves
in a large box but are ignoring the
edges
VM Ayres, ECE802-604, F13
Combine with this idea:
e1
kx
y-axis is E.
The bowl is the KE that an e- has above the minimum requirement of ES
required to be out of a bond
p = hbar k and KE = p2/ 2m
VM Ayres, ECE802-604, F13
Count the number of available energy levels in a 2-DEG
conduction band: NT(E)
VM Ayres, ECE802-604, F13
2-DEG in a semiconductor: NT(E)
VM Ayres, ECE802-604, F13
2-DEG in a semiconductor : NT(E)
VM Ayres, ECE802-604, F13
2-DEG in a semiconductor: NT(E)
VM Ayres, ECE802-604, F13
2-DEG in a semiconductor: NT(E)
VM Ayres, ECE802-604, F13
2-DEG in a semiconductor : NT(E)
VM Ayres, ECE802-604, F13
2-DEG in a semiconductor : NT(E)
VM Ayres, ECE802-604, F13
2-DEG in a semiconductor: NT(E)
VM Ayres, ECE802-604, F13
Use NT(E) to get energy density of states N(E):
VM Ayres, ECE802-604, F13
Your Homework Pr 1.3: 1 Deg in a semiconductor:
VM Ayres, ECE802-604, F13
Your Homework Pr 1.3: 1 Deg in a semiconductor:
VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
Use N(E) to get concentration nS
VM Ayres, ECE802-604, F13
Use N(E) to get concentration nS
VM Ayres, ECE802-604, F13
VM Ayres, ECE802-604, F13
Fermi wavenumber kf:
VM Ayres, ECE802-604, F13
Corresponding Fermi velocityr vf:
VM Ayres, ECE802-604, F13
Characteristic mean free path length Lm:
VM Ayres, ECE802-604, F13
Lecture 04, 10 Sep 13
In Chapter 01 in Datta:
Two dimensional electron gas (2-DEG)
DEG goes down, mobility goes up
Define mobility
Proportional to momentum relaxation time tm
Count carriers nS available for current – Pr. 1.3 (1-DEG)
How nS influences scattering in unexpected ways – Pr 1.1 (2-DEG)
One dimensional electron gas (1-DEG)
Special Schrödinger eqn (Con E) that accommodates:
Electronic confinement: band bending due to space charge
Useful external B-field
Experimental measure for mobility
VM Ayres, ECE802-604, F13