n vs' Ef topic of HW

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Lecture 11 HEMT I-Vs

• n vs. Ef (topic of HW5)
• n vs Vgate
• HEMT I-V

2
HW 5 hints

• 5.12 Use eqns. 5-99, 5-106, 5-107
• 5-13a Use eqns. 5-115,1-90,5-113,5-119
• 5-13b Use eqns. 5-121, 5-131, 5-133. Be sure to
  check whether you are in saturation region.

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2DEG
E0
E0
f2
f1
Ec
Ec
DEc
EFermi
EFermi
GaAs
Ev
AlGaAs
DEv
Ev
4
HEMT
Schottky diode
gate
source
drain
metal (e.g. aluminum)
ohmic
ohmic
n-AlGaAs
tb
i-AlGaAs
d
2DEG
i-GaAs
Insulating substrate
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HEMT
Schottky diode
gate
source
drain
metal (e.g. aluminum)
ohmic
ohmic
depletion region
n-AlGaAs
tb
i-AlGaAs
d
2DEG
i-GaAs
Insulating substrate
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Many possible variations

• Quantum well is also popular.
• Highly doped material under ohmics for low
  contact resistance
• InP based materials
• GaN based materials
• (pHEMT strained materials)

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Band gaps
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2DEG
E0
E0
f2
f1
Ec
Ec
DEc
EFermi
EFermi
GaAs
Ev
AlGaAs
DEv
Ev
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Triangle vs. square well
Ec
Fermi energy
1.4 eV
V
z
Ev
(Draw both bound states on board. In particular
discuss figure 5.21 from Liu.) Also discuss
shallow vs. wide wells on board. (Typically 100
angstroms works.)
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Fermi energy in 2 dimensions
EEFermi
energy
All these states are filled with electrons.
In GaAs, 1011cm-2 gives Ef 1-10 meV But
1012cm-2 gives more than first subband.
E0
Discuss subband, how above integral gets
modified.
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Problem

• Presence of electrons changes shape of potential
  well.
• We need a way to account for this.
• Will do NOW.
• Why? We want to know how many electrons there
  are!
• Later, we want to know how gate voltage changes
  that.

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Trianglular wells
From quantum mechanics
V(x)
x
Ef
Ef
E1
E1
Slope of this line is
Note Lecture 10 had error of hbar.
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2DEG
E0
E0
f2
f1
Ec
Ec
DEc
EFermi
EFermi
GaAs
Ev
AlGaAs
DEv
Ev
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Poisson equation
Actually first of Maxwells four equations
In the x-direction only
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p n diode
-
p type
n type

-

-

-

-

-

no electrons or holes in depletion region
charge density
draw p,n density on board
electric field
potential
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Hint for HW 5.8
n AlGaAs
insulating GaAs
no electrons or holes in depletion region
2DEG
undoped AlGaAs
Nd1
charge density
Xdep,1
d
electric field
Part b, calculate
Then calculate
Then eliminate Xdep,1
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More hints for HW 5.8
So part b and Poisson
relates ns to results of part b.
is a function of the Fermi energy. (You should
write this explicitly in terms of the following
parameters for the HW)
Use figure 5.21 to help you.
Note We have been assuming zero temperature. HW
want you to consider non-zero temperature.
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Trianglular wells
V(x)
x
Ef
E1
Slope of this line is
That gives one relationship between Ef and ns. To
solve for both of them, you need
another relationship HW 5.8
Discuss intuitively adding electrons changes
slope which changes Fermi energy.
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From HW 5.8
You will find
Want to engineer material so that DEc large. For
GaAs, there is a limit. For InxGa1-xAs/InxAl1-xAs,
use strained layers to get larger DEc
(discuss). (InP has higher mobility, peak
velocity than GaAs.)
Called pseudomorphic HEMT pHEMT.
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Band gaps
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ns vs Ef
After all that mumbo-jumbo, we know it is
complicated. We approximate it many times as
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Gate
Gate changes Ef with respect to reference
energy. This changes density.
From Liu
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Vary gate voltage
Changes Fermi energy which changes density.
(Draw better pictures on board.)
From Liu.
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Density
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HEMT analysis
From Liu.
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Density
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Current
J is 2d, ns is 2d. (Discuss).
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Integrating
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Conclusion
IDS
IDS,sat
Previous equation
VDS
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Channel potential
Since we are in 2d, no position dependent
thickness b(x). Life is easier. It can be shown
that
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Velocity saturation

• Just like MESFETs
• Important in short channel HEMTs
• Need to model channel as to regions saturated
  and unsaturated
• Qualitative IVs are similar