ECE 802 604 Nanoelectronics Prof Virginia Ayres Electrical

ECE 802 -604: Nanoelectronics Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University ayresv@msu. edu

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 802 -604, F 13

Lecture 02, 03 Sep 13 Two dimensional electron gas (2 -DEG): Datta example: Ga. As-Al 0. 3 Ga 0. 7 As heterostructure HEMT VM Ayres, ECE 802 -604, F 13

MOSFET Sze VM Ayres, ECE 802 -604, F 13

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 VM Ayres, ECE 802 -604, F 13

For both, the channel is a 2 -DEG that is created electronically by band-bending MOSFET 2 x y. Bp = VM Ayres, ECE 802 -604, F 13

For both, the channel is a 2 -DEG that is created electronically by band-bending HEMT VM Ayres, ECE 802 -604, F 13

Example: Find the correct energy band-bending diagram for a HEMT made from the following heterojunction. p-type Ga. As n-type Al 0. 3 Ga 0. 7 As Heavily doped Moderately doped EC 2 EF 2 EC 1 EF 1 EV 1 Eg 1 = 1. 424 e. V Eg 2 = 1. 798 e. V EV 2 VM Ayres, ECE 802 -604, F 13

p-type Ga. As n-type Al 0. 3 Ga 0. 7 As Heavily doped Moderately doped EC 2 EF 2 EC 1 EF 1 EV 1 Eg 1 = 1. 424 e. V Eg 2 = 1. 798 e. V EV 2 VM Ayres, ECE 802 -604, F 13

Evac qc 1 qfm 1 qc 2 qfm 2 EC 2 EF 2 EC 1 EF 1 EV 1 Evac Eg 1 Eg 2 EV 2 VM Ayres, ECE 802 -604, F 13

Electron affinities qc for Ga. As and Alx. Ga 1 -x. As can be found on Ioffe Evac qc 1 qfm 1 qc 2 qfm 2 EC 2 EF 2 EC 1 EF 1 EV 1 Evac Eg 1 Eg 2 EV 2 VM Ayres, ECE 802 -604, F 13

True for all junctions: align Fermi energy levels: EF 1 = EF 2. This brings Evac along too since electron affinities can’t change VM Ayres, ECE 802 -604, F 13

Put in Junction J, nearer to the more heavily doped side: Junction J VM Ayres, ECE 802 -604, F 13

Join Evac smoothly: J VM Ayres, ECE 802 -604, F 13

Anderson Model: Use qc 1 “measuring stick” to put in EC 1: J VM Ayres, ECE 802 -604, F 13

Use qc 1 “measuring stick” to put in EC 1: J VM Ayres, ECE 802 -604, F 13

Result so far: EC 1 band-bending: J VM Ayres, ECE 802 -604, F 13

Use qc 2 “measuring stick” to put in EC 2: J VM Ayres, ECE 802 -604, F 13

Use qc 2 “measuring stick” to put in EC 2: J VM Ayres, ECE 802 -604, F 13

Results so far: EC 1 and EC 2 band-bending: J VM Ayres, ECE 802 -604, F 13

Put in straight piece connector: J DEC VM Ayres, ECE 802 -604, F 13

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 802 -604, F 13

Use the energy bandgap Eg 1 “measuring stick” to relate EC 1 and EV 1: J DEC VM Ayres, ECE 802 -604, F 13

Use the energy bandgap Eg 1 “measuring stick” to relate EC 1 and EV 1: J DEC VM Ayres, ECE 802 -604, F 13

Result: band-bending for EV 1: J DEC VM Ayres, ECE 802 -604, F 13

Use the energy bandgap Eg 2 “measuring stick” to put in EV 2: J DEC VM Ayres, ECE 802 -604, F 13

Use the energy bandgap Eg 2 “measuring stick” to put in EV 2: J VM Ayres, ECE 802 -604, F 13

Results: band-bending for EV 1 and EV 2: J DEC VM Ayres, ECE 802 -604, F 13

Put in straight piece connector: J DEC DEV Note: for this heterojunction: DEC > DEV VM Ayres, ECE 802 -604, F 13

Put in straight piece connector: J DEC DEV DEC = D(electron affinities) = qc 2 – qc 1 (Anderson model) DEV = ( E 2 – E 1 ) - DEC => DEgap = DEC + DEV VM Ayres, ECE 802 -604, F 13

Put in straight piece connector: J DEC DEV “The difference in the energy bandgaps is accommodated by amount DEC in the conduction band amount DEV in the valence band. ” VM Ayres, ECE 802 -604, F 13

J DEC DEV NO quantum well in EV NO quantum well for holes VM Ayres, ECE 802 -604, F 13

Correct band-bending diagram: J DEC DEV VM Ayres, ECE 802 -604, F 13

Is the Example the same as the example in Datta? HEMT VM Ayres, ECE 802 -604, F 13

No. The L-R orientation is trivial but the starting materials are different Our example Datta example VM Ayres, ECE 802 -604, F 13

Orientation is trivial. The smaller bandgap material is always “ 1” Our example Datta example VM Ayres, ECE 802 -604, F 13

HEMT Physical region In this region: a triangular quantum well has developed in the conduction band. 2 -DEG Allowed energy levels VM Ayres, ECE 802 -604, F 13

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 802 -604, F 13

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 802 -604, F 13

Example: approximate the real well by a one dimensional triangular well in z ∞ Using information from ECE 874 Pierret problem 2. 7 (next page), evaluate the quantized part of the energy of an electron that occupies the 1 st energy level VM Ayres, ECE 802 -604, F 13

VM Ayres, ECE 802 -604, F 13

U(z) = az z VM Ayres, ECE 802 -604, F 13

n=? m=? a=? VM Ayres, ECE 802 -604, F 13

n = 0 for 1 st m = meff for conduction band e- in Ga. As. At 300 K this is 0. 067 m 0 a=? VM Ayres, ECE 802 -604, F 13

Your model for a = asymmetry ? U(z) = 3/2 z U(z) = 1 z z VM Ayres, ECE 802 -604, F 13

D. L. Mathine, G. N. Maracas, D. S. Gerber, R. Droopad, R. J. Graham, and M. R. Mc. Cartney. Characterization of an Al. Ga. As/Ga. As 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 Ben. Daniel–Duke Hamiltonian [D. J. Ben. Daniel 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 E 11 h, E 11 l, and E 22 l 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 802 -604, F 13

Jin Xiao (金� ), Zhang Hong (�� ), Zhou Rongxiu (周荣秀) and Jin Zhao (金� ). Interface roughness scattering in an Al. Ga. As/Ga. As 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 (2 DEG) 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 Alx. Ga 1−x. As/Ga. As 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 802 -604, F 13