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**ECE 802-604: Nanoelectronics**

Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University

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**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 VM Ayres, ECE , F13

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**Lecture 02, 03 Sep 13 Two dimensional electron gas (2-DEG):**

Datta example: GaAs-Al0.3Ga0.7As heterostructure HEMT VM Ayres, ECE , F13

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MOSFET Sze VM Ayres, ECE , F13

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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 VM Ayres, ECE , F13

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

MOSFET 2 x yBp = VM Ayres, ECE , F13

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

HEMT VM Ayres, ECE , F13

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Example: Find the correct energy band-bending diagram for a HEMT made from the following heterojunction. p-type GaAs Heavily doped n-type Al0.3Ga0.7As Moderately doped EC2 EF2 EC1 Eg1 = eV Eg2 = eV EF1 EV1 EV2 VM Ayres, ECE , F13

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**p-type GaAs Heavily doped n-type Al0.3Ga0.7As Moderately doped**

EC2 EF2 EC1 Eg1 = eV Eg2 = eV EF1 EV1 EV2 VM Ayres, ECE , F13

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**Evac Evac qc1 qfm1 qc2 qfm2 EC2 EF2 EC1 Eg1 Eg2 EF1 EV1 EV2**

VM Ayres, ECE , F13

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

Evac Evac qc1 qfm1 qc2 qfm2 EC2 EF2 EC1 Eg1 Eg2 EF1 EV1 EV2 VM Ayres, ECE , F13

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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 VM Ayres, ECE , F13

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

VM Ayres, ECE , F13

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Join Evac smoothly: J VM Ayres, ECE , F13

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**Anderson Model: Use qc1 “measuring stick” to put in EC1:**

J VM Ayres, ECE , F13

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**Use qc1 “measuring stick” to put in EC1:**

J VM Ayres, ECE , F13

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**Result so far: EC1 band-bending:**

J VM Ayres, ECE , F13

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**Use qc2 “measuring stick” to put in EC2:**

J VM Ayres, ECE , F13

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**Use qc2 “measuring stick” to put in EC2:**

J VM Ayres, ECE , F13

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**Results so far: EC1 and EC2 band-bending:**

J VM Ayres, ECE , F13

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**Put in straight piece connector:**

J DEC VM Ayres, ECE , F13

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**(for this heterojunction combination):**

Keeping the electron affinities correct resulted in a triangular quantum well in EC (for this heterojunction combination): J DEC In this region: a triangular quantum well has developed in the conduction band VM Ayres, ECE , F13

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

J DEC VM Ayres, ECE , F13

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

J DEC VM Ayres, ECE , F13

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**Result: band-bending for EV1:**

J DEC VM Ayres, ECE , F13

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

J DEC VM Ayres, ECE , F13

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

J VM Ayres, ECE , F13

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**Results: band-bending for EV1 and EV2:**

J DEC VM Ayres, ECE , F13

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**Put in straight piece connector:**

J DEC DEV Note: for this heterojunction: DEC > DEV VM Ayres, ECE , F13

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**Put in straight piece connector:**

J DEC DEV DEC = D(electron affinities) = qc2 – qc1 (Anderson model) DEV = ( E2 – E1 ) - DEC => DEgap = DEC + DEV VM Ayres, ECE , F13

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**Put in straight piece connector:**

J DEC DEV “The difference in the energy bandgaps is accommodated by amount DEC in the conduction band and amount DEV in the valence band.” VM Ayres, ECE , F13

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**NO quantum well for holes**

J DEC DEV NO quantum well in EV NO quantum well for holes VM Ayres, ECE , F13

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**Correct band-bending diagram:**

J DEC DEV VM Ayres, ECE , F13

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**Is the Example the same as the example in Datta?**

HEMT VM Ayres, ECE , F13

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

Our example Datta example VM Ayres, ECE , F13

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

Our example Datta example VM Ayres, ECE , F13

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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 VM Ayres, ECE , F13

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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 VM Ayres, ECE , F13

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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 VM Ayres, ECE , F13

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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 VM Ayres, ECE , F13

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VM Ayres, ECE , F13

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U(z) = az z VM Ayres, ECE , F13

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n = ? m = ? a = ? VM Ayres, ECE , F13

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

n = 0 for 1st m = meff for conduction band e- in GaAs. At 300K this is m0 a = ? VM Ayres, ECE , F13

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**Your model for a = asymmetry ?**

U(z) = 3/2 z U(z) = 1 z z VM Ayres, ECE , F13

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**D. L. Mathine, G. N. Maracas, D. S. Gerber, R. Droopad, R. J**

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 VM Ayres, ECE , F13

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**Jin Xiao (金晓), Zhang Hong (张红), Zhou Rongxiu (周荣秀) and Jin Zhao (金钊)**

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 , 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 VM Ayres, ECE , F13

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