ece 802-604: nanoelectronics prof. virginia ayres electrical & computer engineering michigan...

ECE 802-604:Nanoelectronics

Prof. Virginia AyresElectrical & Computer EngineeringMichigan State [email protected]

VM Ayres, ECE802-604, F13

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-DEGUseful external B-field

Experimental measure for mobility


Lecture 02, 03 Sep 13

Two dimensional electron gas (2-DEG):

Datta example: GaAs-Al0.3Ga0.7As heterostructure HEMT


Sze

MOSFET


IOP Science website;Tunnelling- and barrier-injection transit-time mechanisms of terahertz plasma instability in high-electron mobility transistors2002 Semicond. Sci. Technol. 17 1168

HEMT

http://iopscience.iop.org/0268-1242/17/11/306

http://iopscience.iop.org/0268-1242/17/11/306


For both, the channel is a 2-DEG that is created electronically by band-bending

MOSFET

2 x Bp =


HEMT

For both, the channel is a 2-DEG that is created electronically by band-bending


= 1.798 eV= 1.424 eVEg1

EC1

EF1EV1

EC2EF2

EV2

Eg2

p-type GaAs

Heavily doped

n-type Al0.3Ga0.7As

Moderately doped

Example: Find the correct energy band-bending diagram for a HEMT made from the following heterojunction.


= 1.798 eV= 1.424 eVEg1

EC1

EF1EV1

EC2EF2

EV2

Eg2

p-type GaAs

Heavily doped

n-type Al0.3Ga0.7As

Moderately doped


Eg1

EC1

Evac

qm1

EF1EV1

q1

Evac

qm2q2

EC2EF2

EV2

Eg2


Eg1

EC1

Evac

qm1

EF1EV1

q1

Evac

qm2q2

EC2EF2

EV2

Eg2

Electron affinities q for GaAs and AlxGa1-xAs can be found on Ioffe


True for all junctions: align Fermi energy levels: EF1 = EF2.This brings Evac along too since electron affinities can’t change

Eg1

EC1

Evac

qm1

EF1EV1

q1Evac

qm2q2

EC2EF2

EV2

Eg2


Put in Junction J, nearer to the more heavily doped side:

Junction J

Evac

qm2q2

EC2EF2

EV2

Eg2

Eg1

EC1

Evac

qm1

EF1EV1

q1


Evac

qm2q2

EC2EF2

EV2

Eg2

Join Evac smoothly:

J

Eg1

EC1

Evac

qm1

EF1EV1

q1


Evac

qm2q2

EC2EF2

EV2

Eg2

Eg1

EC1

Evac

qm1

EF1EV1

q1

Anderson Model: Use q1 “measuring stick” to put in EC1:

J


Evac

qm2q2

EC2EF2

EV2

Eg2

Eg1

EC1

Evac

qm1

EF1EV1

q1

Use q1 “measuring stick” to put in EC1:

J


Evac

qm2q2

EC2EF2

EV2

Eg2

Eg1

EC1

Evac

qm1

EF1EV1

q1

Result so far: EC1 band-bending:

J


Evac

qm2q2

EC2EF2

EV2

Eg2

Eg1

EC1

Evac

qm1

EF1EV1

q1

Use q2 “measuring stick” to put in EC2:

J


Evac

qm2q2

EC2EF2

EV2

Eg2

Eg1

EC1

Evac

qm1

EF1EV1

q1

Results so far: EC1 and EC2 band-bending:

J


Evac

qm2q2

EC2EF2

EV2

Eg2

Eg1

EC1

Evac

qm1

EF1EV1

q1

Put in straight piece connector:

J

EC


Evac

qm2q2

EC2EF2

EV2

Eg2

Eg1

EC1

Evac

qm1

EF1EV1

q1

Keeping the electron affinities correct resulted in a triangular quantum well in EC (for this heterojunction combination):

J

In this region: a triangular quantum well has developed in the conduction band

EC


Evac

qm2q2

EC2EF2

EV2

Eg2

Eg1

EC1

Evac

qm1

EF1EV1

q1

Use the energy bandgap Eg1 “measuring stick” to relate EC1 and EV1:

J

EC


Evac

qm2q2

EC2EF2

EV2

Eg2

Eg1

EC1

Evac

qm1

EF1EV1

q1

Result: band-bending for EV1:

J

EC


Evac

qm2q2

EC2EF2

EV2

Eg2

Eg1

EC1

Evac

qm1

EF1EV1

q1

Use the energy bandgap Eg2 “measuring stick” to put in EV2:

J

EC


Evac

qm2q2

EC2EF2

EV2

Eg2

Eg1

EC1

Evac

qm1

EF1EV1

q1

Use the energy bandgap Eg2 “measuring stick” to put in EV2:

J


Evac

qm2q2

EC2EF2

EV2

Eg2

Eg1

EC1

Evac

qm1

EF1EV1

q1

Results: band-bending for EV1 and EV2:

J

EC


Evac

qm2q2

EC2EF2

EV2

Eg2

Eg1

EC1

Evac

qm1

EF1EV1

q1


J

Note: for this heterojunction:EC > EV

EC

EV


Evac

qm2q2

EC2EF2

EV2

Eg2

Eg1

EC1

Evac

qm1

EF1EV1

q1


J

EC

EV

EC = (electron affinities) = q2 – q1

(Anderson model)

EV = ( E2 – E1 ) - EC => Egap = EC + EV


Evac

qm2q2

EC2EF2

EV2

Eg2

Eg1

EC1

Evac

qm1

EF1EV1

q1


J

EC

EV

“The difference in the energy bandgaps is accommodated by amount EC in the conduction band and amount EV in the valence band.”


Evac

qm2q2

EC2EF2

EV2

Eg2

Eg1

EC1

Evac

qm1

EF1EV1

q1

J

NO quantum well in EV

NO quantum well for holes

EC

EV


Evac

qm2q2

EC2EF2

EV2

Eg2

Eg1

EC1

Evac

qm1

EF1EV1

q1

Correct band-bending diagram:

J

EC

EV


HEMT

Is the Example the same as the example in Datta?


No. The L-R orientation is trivial but the starting materials are different

= 1.798 eV = 1.424 eVEg1

EC1

EF1EV1

EC2EF2

EV2

Eg2

p-type GaAs

Heavily doped

n-type Al0.3Ga0.7As

Moderately doped

= 1.798 eV = 1.424 eVEg1

EC1EF1

EV1

EC2EF2

EV2

Eg2

intrinsic GaAs

undoped

n-type Al0.3Ga0.7As

Moderately doped

Our example

Datta example


Orientation is trivial. The smaller bandgap material is always “1”

= 1.798 eV = 1.424 eVEg1

EC1

EF1EV1

EC2EF2

EV2

Eg2

p-type GaAs

Heavily doped

n-type Al0.3Ga0.7As

Moderately doped

= 1.798 eV = 1.424 eVEg1

EC1EF1

EV1

EC2EF2

EV2

Eg2

intrinsic GaAs

undoped

n-type Al0.3Ga0.7As

Moderately doped

Our example

Datta example


HEMT

In this region: a triangular quantum well has developed in the conduction band.2-DEG Allowed energy levels

Physical region


Example: Which dimension (axis) is quantized? zWhich dimensions form the 2-DEG? x and y


Physical region


Example: Which dimension is quantized?Which dimensions form the 2-DEG?


Physical region


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


U(z) = az

z


n = ?

m = ?

a = ?


n = 0 for 1st

m = meff for conduction band e- in GaAs. At 300K this is 0.067 m0

a = ?


Your model for a = asymmetry ?

∞

z

U(z) = 3/2 z

U(z) = 1 z


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


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

ece 802-604: nanoelectronics prof. virginia ayres electrical & computer engineering michigan...

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