ece 802-604: nanoelectronics
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ECE 802-604: Nanoelectronics. Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University [email protected]. Lecture 04, 10 Sep 13. In Chapter 01 in Datta: Two dimensional electron gas (2-DEG) DEG goes down, mobility goes up Define mobility - PowerPoint PPT PresentationTRANSCRIPT
ECE 802-604:Nanoelectronics
Prof. Virginia AyresElectrical & Computer EngineeringMichigan State [email protected]
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 m
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 chargeUseful 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 Proportional to momentum relaxation time m
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 chargeUseful external B-field
Experimental measure for mobility
VM Ayres, ECE802-604, F13
z
y
x
-z
y
Wire up HEMT to use the triangular quantum well region in GaAs
Correct for e-’s with Drain = +Note: current I is IDS
n-E y
= (-|e |)(-|E y|)
VM Ayres, ECE802-604, F13
Why do this: increase in Mobility in using 2-DEG region in GaAs instead of 3-DEG bulk GaAs
931C: 3D Scattering
T = hot:Phonon latticescattering
T = cold:Impurity = ND+, NA- scattering
Sweet spot at 300K
mo
bil
ity
VM Ayres, ECE802-604, F13
Scattering involves energy and momentum conserving interactions. Putting quantum restrictions on these interactions means that fewer can occur.
Increase in Mobility is based on decrease of scattering, or said another way, increase e-s not scattered.
VM Ayres, ECE802-604, F13
Streetman t:
Datta m t: The statement below is true for a group of e-s not a single scattering event. m 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 Proportional to momentum relaxation time m
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 chargeUseful external B-field
Experimental measure for mobility
VM Ayres, ECE802-604, F13
2-DEG: Major improvement in performance at low temperatures
931C: 3D Scattering
T = hot:Phonon latticescattering
T = cold:Impurity = ND+, NA- scattering
Sweet spot at 300K
mo
bil
ity
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:
3-DEG
intrinisic
VM Ayres, ECE802-604, F13
2-DEG: Energy:
Special Schrödinger eqn (Con E) that accommodates:Electronic confinement: band bending due to space chargeUseful 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
Macroscopic World Bulk Materials
y
x
z
px2 + py
2 + pz2
2m* 2m* 2m*KE =
Silicon Ingot
Free motion in all directions
px , py , pz can take any values
B. Jacobs, PhD thesis
VM Ayres, ECE802-604, F13
Thin Films
y
x
z
px2 + py
2 + nz2 ħ22
2m* 2m* 2m*Lz2
J.S. Moodera, Francis Bitter Magnet Lab, MITA.K. Geim and K.S. Novoselov, Nat. Mater., 2007, 6, 183
Graphene
Free motion in x and y directions
Shown: Infinite potential well in z direction
pz is constrained to be a number(s)
Thin layers
Reduced Dimensionality Systems: 2-DEG
E =
KE
B. Jacobs, PhD thesis
VM Ayres, ECE802-604, F13
Carbon Nanotubes,
Nanowires,
Molecular Electronics
y
x
z nx
2 ħ22 + py2 + nz
2 ħ22
2m*Lx2 2m* 2m*Lz
2
Richard E. Smalley Institute, Rice University
1μm
Reduced Dimensionality Systems: 1-DEG
Free motion in y direction
Shown: Infinite potential well in x and z directions
px , pz are constrained to be a number(s)
E =
KE
B. Jacobs, PhD thesis
VM Ayres, ECE802-604, F13
Quantum Dots
z
yx
nx2 ħ22 + ny
2 ħ22 + nz2 ħ22
2m*Lx2 2m*Ly
2 2m*Lz2
A. Kadavanich, MRSCE, University of Wisconsin
Reduced Dimensionality Systems: 0-DEG
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)
E =
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:
1
Similar to:
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:
1
VM Ayres, ECE802-604, F13
2-DEG in a semiconductor:
1
kx
Any little patch on there would have some values of kx, ky
VM Ayres, ECE802-604, F13
2-DEG in a semiconductor:
1
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
Thin Films
y
x
z
px2 + py
2 + nz2 ħ22
2m* 2m* 2m*Lz2
J.S. Moodera, Francis Bitter Magnet Lab, MITA.K. Geim and K.S. Novoselov, Nat. Mater., 2007, 6, 183
Graphene
Free motion in x and y directions
Shown: Infinite potential well in z direction
pz is constrained to be a number(s)
Thin layers
Reduced Dimensionality Systems: 2-DEG
E =
KE: write p in terms of hbar k
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:
1
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 m
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 chargeUseful external B-field
Experimental measure for mobility