Transcript Lecture 02 03 Sep 13 - College of Engineering, Michigan State

ECE 802-604:
Nanoelectronics
Prof. Virginia Ayres
Electrical & Computer Engineering
Michigan State University
[email protected]
Lecture 02, 03 Sep 13
In Chapter 01 in Datta:
Two dimensional electron gas (2-DEG)
DEG goes down, mobility goes up
Define mobility (and momentum relaxation)
One dimensional electron gas (1-DEG)
Special Schrödinger eqn (Con E) that accommodates:
Confinement to create 1-DEG
Useful external B-field
Experimental measure for mobility
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Lecture 02, 03 Sep 13
Two dimensional electron gas (2-DEG):
Datta example: GaAs-Al0.3Ga0.7As heterostructure HEMT
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MOSFET
Sze
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HEMT
IOP Science website;
Tunnelling- and barrier-injection transit-time mechanisms of terahertz
plasma instability in high-electron mobility transistors
2002 Semicond. Sci. Technol. 17 1168
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For both, the channel is a 2-DEG that is created electronically
by band-bending
MOSFET
2 x yBp =
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For both, the channel is a 2-DEG that is created electronically
by band-bending
HEMT
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Example: Find the correct energy band-bending diagram for a
HEMT made from the following heterojunction.
p-type GaAs
n-type Al0.3Ga0.7As
Heavily doped
Moderately doped
EC2
EF2
EC1
EF1
EV1
Eg1 = 1.424 eV
Eg2 = 1.798 eV
EV2
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p-type GaAs
n-type Al0.3Ga0.7As
Heavily doped
Moderately doped
EC2
EF2
EC1
EF1
EV1
Eg1 = 1.424 eV
Eg2 = 1.798 eV
EV2
VM Ayres, ECE802-604, F13
Evac
qc1
qfm1
qc2
qfm2
EC2
EF2
EC1
EF1
EV1
Evac
Eg1
Eg2
EV2
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Electron affinities qc for GaAs and AlxGa1-xAs can be found on Ioffe
Evac
qc1
qfm1
qc2
qfm2
EC2
EF2
EC1
EF1
EV1
Evac
Eg1
Eg2
EV2
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True for all junctions: align Fermi energy levels: EF1 = EF2.
This brings Evac along too since electron affinities can’t change
Evac
qc1
Evac
qfm1
qc2
qfm2
EC1
EF1
EV1
Eg1
EC2
EF2
Eg2
EV2
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Put in Junction J, nearer to the more heavily doped side:
Evac
qc1
Junction J
Evac
qfm1
qc2
qfm2
EC1
EF1
EV1
Eg1
EC2
EF2
Eg2
EV2
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Join Evac smoothly:
Evac
qc1
J
Evac
qfm1
qc2
qfm2
EC1
EF1
EV1
Eg1
EC2
EF2
Eg2
EV2
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Anderson Model: Use qc1 “measuring stick” to put in EC1:
Evac
qc1
J
Evac
qfm1
qc2
qfm2
EC1
EF1
EV1
Eg1
EC2
EF2
Eg2
EV2
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Use qc1 “measuring stick” to put in EC1:
Evac
qc1
J
Evac
qfm1
qc2
qfm2
EC1
EF1
EV1
Eg1
EC2
EF2
Eg2
EV2
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Result so far: EC1 band-bending:
Evac
qc1
J
Evac
qfm1
qc2
qfm2
EC1
EF1
EV1
Eg1
EC2
EF2
Eg2
EV2
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Use qc2 “measuring stick” to put in EC2:
Evac
qc1
J
Evac
qfm1
qc2
qfm2
EC1
EF1
EV1
Eg1
EC2
EF2
Eg2
EV2
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Use qc2 “measuring stick” to put in EC2:
Evac
qc1
J
Evac
qfm1
qc2
qfm2
EC1
EF1
EV1
Eg1
EC2
EF2
Eg2
EV2
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Results so far: EC1 and EC2 band-bending:
Evac
qc1
J
Evac
qfm1
qc2
qfm2
EC1
EF1
EV1
Eg1
EC2
EF2
Eg2
EV2
VM Ayres, ECE802-604, F13
Put in straight piece connector:
Evac
qc1
J
Evac
qfm1
qc2
DEC
EC1
EF1
EV1
qfm2
Eg1
EC2
EF2
Eg2
EV2
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Keeping the electron affinities correct resulted in a triangular quantum well in EC
(for this heterojunction combination):
J
Evac
qc1
Evac
qfm1
qc2
DEC
EC1
EF1
EV1
qfm2
Eg1
EC2
EF2
Eg2
In this region: a triangular
quantum well has developed
in the conduction band
EV2
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Use the energy bandgap Eg1 “measuring stick” to relate EC1 and EV1:
Evac
qc1
J
Evac
qfm1
qc2
DEC
EC1
EF1
EV1
qfm2
Eg1
EC2
EF2
Eg2
EV2
VM Ayres, ECE802-604, F13
Use the energy bandgap Eg1 “measuring stick” to relate EC1 and EV1:
Evac
qc1
J
Evac
qfm1
qc2
DEC
EC1
EF1
EV1
qfm2
Eg1
EC2
EF2
Eg2
EV2
VM Ayres, ECE802-604, F13
Result: band-bending for EV1:
Evac
qc1
J
Evac
qfm1
qc2
DEC
EC1
EF1
EV1
qfm2
Eg1
EC2
EF2
Eg2
EV2
VM Ayres, ECE802-604, F13
Use the energy bandgap Eg2 “measuring stick” to put in EV2:
Evac
qc1
J
Evac
qfm1
qc2
DEC
EC1
EF1
EV1
qfm2
Eg1
EC2
EF2
Eg2
EV2
VM Ayres, ECE802-604, F13
Use the energy bandgap Eg2 “measuring stick” to put in EV2:
Evac
qc1
J
Evac
qfm1
qc2
qfm2
EC1
EF1
EV1
Eg1
EC2
EF2
Eg2
EV2
VM Ayres, ECE802-604, F13
Results: band-bending for EV1 and EV2:
Evac
qc1
J
Evac
qfm1
qc2
DEC
EC1
EF1
EV1
qfm2
Eg1
EC2
EF2
Eg2
EV2
VM Ayres, ECE802-604, F13
Put in straight piece connector:
Evac
qc1
J
Evac
qfm1
qc2
DEC
EC1
EF1
EV1
qfm2
Eg1
EC2
EF2
DEV
Eg2
Note: for this heterojunction:
DEC > DEV
EV2
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Put in straight piece connector:
Evac
qc1
J
Evac
qfm1
qc2
DEC
EC1
EF1
EV1
qfm2
Eg1
EC2
EF2
DEV
DEC = D(electron affinities) = qc2 – qc1
(Anderson model)
DEV = ( E2 – E1 ) - DEC => DEgap = DEC + DEV
Eg2
EV2
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Put in straight piece connector:
Evac
qc1
J
Evac
qfm1
qc2
DEC
EC1
EF1
EV1
qfm2
Eg1
EC2
EF2
DEV
“The difference in the energy bandgaps
is accommodated by amount DEC in the
conduction band and amount DEV in
the valence band.”
Eg2
EV2
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Evac
qc1
J
Evac
qfm1
qc2
DEC
EC1
EF1
EV1
qfm2
Eg1
EC2
EF2
DEV
Eg2
NO quantum well in EV
NO quantum well for holes
EV2
VM Ayres, ECE802-604, F13
Correct band-bending diagram:
Evac
qc1
J
Evac
qfm1
qc2
DEC
EC1
EF1
EV1
qfm2
Eg1
EC2
EF2
DEV
Eg2
EV2
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Is the Example the same as the example in Datta?
HEMT
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No. The L-R orientation is trivial but the starting materials are different
n-type Al0.3Ga0.7As
p-type GaAs
Moderately doped
Heavily doped
EC2
EF2
Our example
Eg2 = 1.798 eV
EC1
Eg1 = 1.424 eV
EF1
EV1
EV2
n-type Al0.3Ga0.7As
intrinsic GaAs
Moderately doped
undoped
EC2
EF2
Datta example
Eg2 = 1.798 eV
EC1
EF1
Eg1 = 1.424 eV
EV1
EV2
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Orientation is trivial. The smaller bandgap material is always “1”
n-type Al0.3Ga0.7As
p-type GaAs
Moderately doped
Heavily doped
EC2
EF2
Our example
Eg2 = 1.798 eV
EC1
Eg1 = 1.424 eV
EF1
EV1
EV2
n-type Al0.3Ga0.7As
intrinsic GaAs
Moderately doped
undoped
EC2
EF2
Datta example
Eg2 = 1.798 eV
EC1
EF1
Eg1 = 1.424 eV
EV1
EV2
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HEMT
Physical region
In this region: a triangular quantum
well has developed in the conduction
band.
2-DEG Allowed energy levels
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Example:
Which dimension (axis) is quantized? z
Which dimensions form the 2-DEG? x and y
Physical region
In this region: a triangular quantum
well has developed in the conduction
band.
2-DEG Allowed energy levels
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Example:
Which dimension is quantized?
Which dimensions form the 2-DEG?
Physical region
In this region: a triangular quantum
well has developed in the conduction
band.
2-DEG Allowed energy levels
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Example: approximate the real well by a one dimensional
triangular well in z
∞
Using information from ECE874 Pierret problem 2.7 (next page),
evaluate the quantized part of the energy of an electron that
occupies the 1st energy level
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U(z) = az
z
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n=?
m=?
a=?
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n = 0 for 1st
m = meff for conduction band e- in GaAs. At 300K this is 0.067 m0
a=?
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Your model for a = asymmetry ?
∞
U(z) = 3/2 z
U(z) = 1 z
z
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D. L. Mathine, G. N. Maracas, D. S. Gerber, R. Droopad, R. J.
Graham, and M. R. McCartney. Characterization of an
AlGaAs/GaAs asymmetric triangular quantum well grown by a
digital alloy approximation. J. Appl. Phys. 75, 4551 (1994)
An asymmetric triangular quantum well was grown by
molecular‐beam epitaxy using a digital alloy composition
grading method. A high‐resolution electron micrograph (HREM),
a computational model, and room‐temperature
photoluminescence were used to extract the spatial
compositional dependence of the quantum well. The HREM
micrograph intensity profile was used to determine the shape of
the quantum well. A Fourier series method for solving the
BenDaniel–Duke Hamiltonian [D. J. BenDaniel and C. B. Duke,
Phys. Rev. 152, 683 (1966)] was then used to calculate the
bound energy states within the envelope function scheme for
the measured well shape. These calculations were compared to
the E11h, E11l, and E22l transitions in the room‐temperature
photoluminescence and provided a self‐consistent
compositional profile for the quantum well. A comparison of
energy levels with a linearly graded well is also presented
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Jin Xiao (金晓), Zhang Hong (张红), Zhou Rongxiu (周荣秀) and
Jin Zhao (金钊). Interface roughness scattering in an
AlGaAs/GaAs triangle quantum well and square quantum well.
Journal of Semiconductors Volume 34 072004, 2013
We have theoretically studied the mobility limited by interface
roughness scattering on two-dimensional electrons gas (2DEG)
at a single heterointerface (triangle-shaped quantum well). Our
results indicate that, like the interface roughness scattering in a
square quantum well, the roughness scattering at the
AlxGa1−xAs/GaAs heterointerface can be characterized by
parameters of roughness height Δ and lateral Λ, and in addition
by electric field F. A comparison of two mobilities limited by the
interface roughness scattering between the present result and a
square well in the same condition is given
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