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

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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-DEG Useful external B-field Experimental measure for mobility

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VM Ayres, ECE802-604, F13 Lecture 02, 03 Sep 13 Two dimensional electron gas (2-DEG): Datta example: GaAs-Al 0.3 Ga 0.7 As heterostructure HEMT

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VM Ayres, ECE802-604, F13 Sze MOSFET

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VM Ayres, ECE802-604, F13 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 HEMT

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VM Ayres, ECE802-604, F13 For both, the channel is a 2-DEG that is created electronically by band-bending MOSFET 2 x Bp =

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VM Ayres, ECE802-604, F13 HEMT For both, the channel is a 2-DEG that is created electronically by band-bending

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VM Ayres, ECE802-604, F13 = 1.798 eV = 1.424 eV E g1 E C1 E F1 E V1 E C2 E F2 E V2 E g2 p-type GaAs Heavily doped n-type Al 0.3 Ga 0.7 As Moderately doped Example: Find the correct energy band-bending diagram for a HEMT made from the following heterojunction.

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VM Ayres, ECE802-604, F13 = 1.798 eV = 1.424 eV E g1 E C1 E F1 E V1 E C2 E F2 E V2 E g2 p-type GaAs Heavily doped n-type Al 0.3 Ga 0.7 As Moderately doped

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VM Ayres, ECE802-604, F13 E g1 E C1 E vac q m1 E F1 E V1 q1q1 E vac q m2 q2q2 E C2 E F2 E V2 E g2

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VM Ayres, ECE802-604, F13 E g1 E C1 E vac q m1 E F1 E V1 q1q1 E vac q m2 q2q2 E C2 E F2 E V2 E g2 Electron affinities q for GaAs and Al x Ga 1-x As can be found on Ioffe

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VM Ayres, ECE802-604, F13 True for all junctions: align Fermi energy levels: E F1 = E F2. This brings E vac along too since electron affinities can’t change

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VM Ayres, ECE802-604, F13 Put in Junction J, nearer to the more heavily doped side: Junction J

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VM Ayres, ECE802-604, F13 Join E vac smoothly: J

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VM Ayres, ECE802-604, F13 Anderson Model: Use q 1 “measuring stick” to put in E C1 : J

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VM Ayres, ECE802-604, F13 Use q 1 “measuring stick” to put in E C1 : J

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VM Ayres, ECE802-604, F13 Result so far: E C1 band-bending: J

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VM Ayres, ECE802-604, F13 Use q 2 “measuring stick” to put in E C2 : J

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VM Ayres, ECE802-604, F13 Use q 2 “measuring stick” to put in E C2 : J

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VM Ayres, ECE802-604, F13 Results so far: E C1 and E C2 band-bending: J

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VM Ayres, ECE802-604, F13 Put in straight piece connector: J ECEC

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VM Ayres, ECE802-604, F13 Keeping the electron affinities correct resulted in a triangular quantum well in E C (for this heterojunction combination): J In this region: a triangular quantum well has developed in the conduction band ECEC

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VM Ayres, ECE802-604, F13 Use the energy bandgap E g1 “measuring stick” to relate E C1 and E V1 : J ECEC

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VM Ayres, ECE802-604, F13 Use the energy bandgap E g1 “measuring stick” to relate E C1 and E V1 : J ECEC

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VM Ayres, ECE802-604, F13 Result: band-bending for E V1 : J ECEC

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VM Ayres, ECE802-604, F13 Use the energy bandgap E g2 “measuring stick” to put in E V2 : J ECEC

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VM Ayres, ECE802-604, F13 Use the energy bandgap E g2 “measuring stick” to put in E V2 : J

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VM Ayres, ECE802-604, F13 Results: band-bending for E V1 and E V2 : J ECEC

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VM Ayres, ECE802-604, F13 Put in straight piece connector: J Note: for this heterojunction: E C > E V ECEC EVEV

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VM Ayres, ECE802-604, F13 Put in straight piece connector: J ECEC EVEV E C = (electron affinities) = q 2 – q 1 (Anderson model) E V = ( E 2 – E 1 ) - E C => E gap = E C + E V

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VM Ayres, ECE802-604, F13 Put in straight piece connector: J ECEC EVEV “The difference in the energy bandgaps is accommodated by amount E C in the conduction band and amount E V in the valence band.”

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VM Ayres, ECE802-604, F13 J NO quantum well in E V NO quantum well for holes ECEC EVEV

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VM Ayres, ECE802-604, F13 Correct band-bending diagram: J ECEC EVEV

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VM Ayres, ECE802-604, F13 HEMT Is the Example the same as the example in Datta?

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VM Ayres, ECE802-604, F13 No. The L-R orientation is trivial but the starting materials are different Our example Datta example

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VM Ayres, ECE802-604, F13 Orientation is trivial. The smaller bandgap material is always “1” Our example Datta example

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VM Ayres, ECE802-604, F13 HEMT In this region: a triangular quantum well has developed in the conduction band. 2-DEG Allowed energy levels Physical region

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VM Ayres, ECE802-604, F13 Example: Which dimension (axis) is quantized? z Which dimensions form the 2-DEG? x and y In this region: a triangular quantum well has developed in the conduction band. 2-DEG Allowed energy levels Physical region

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VM Ayres, ECE802-604, F13 Example: Which dimension is quantized? Which dimensions form the 2-DEG? In this region: a triangular quantum well has developed in the conduction band. 2-DEG Allowed energy levels Physical region

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VM Ayres, ECE802-604, F13 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 1 st energy level

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VM Ayres, ECE802-604, F13

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U(z) = az z

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VM Ayres, ECE802-604, F13 n = ? m = ? a = ?

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VM Ayres, ECE802-604, F13 n = 0 for 1st m = m eff for conduction band e- in GaAs. At 300K this is 0.067 m 0 a = ?

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VM Ayres, ECE802-604, F13 Your model for a = asymmetry ? z U(z) = 3/2 z U(z) = 1 z

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VM Ayres, ECE802-604, F13 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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VM Ayres, ECE802-604, F13 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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