Lecture 06

Lecture 06 - David K. Ferry Regents Professor Arizona State...

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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L6:1 EEE 434/591—Quantum Mechanics David K. Ferry Regents’ Professor Arizona State University Royal Palace, Stockholm
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L6:2 Dirac, Heisenberg, and Schrödinger at the time of the award of the 1931 and 1932 Nobel prizes.
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L6:3 Today, we want to discuss triangular potential wells, which occur e.g. in the MOSFET.
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L6:4 The Metal-Oxide-Semiconductor Devices INSULATING OXIDE LAYER n n p-TYPE LAYER CONTROL GATE SOURCE DRAIN n-type contacts are created in the p-type substrate. The n-type channel is INDUCED by the GATE, and connects these two contacts. The number of electrons in the CHANNEL is determined by the GATE, so the CONDUCTANCE is modulated by the GATE.
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L6:5 Metal E F SiO 2 E c E v eV a + - p-type semiconductor This is our triangular well, where the inversion layer electrons reside. Classically, the electron density falls off from the surface as so that the density falls off over a distance corresonding to a drop in potential of k B T .
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L6:6 If we have an inversion density of ~ 5 × 10 12 cm -2 , then Gauss’ Law tells us that the surface field is approximately Then, the distance over which the density decays by 1/ e is approximately On the other hand, the deBroglie wavelength at an energy of k B T is These numbers are incompatible, and we have to solve Schrödinger’s equation in this triangular well.
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L6:7 x V ( x ) = →∞ , x < 0, eF s x , x > 0. Χ Ψ Ω ψ i (0) = 0 i ( x Ei ) x Ei = E i eF s , x 0 = E eF s x Ei E i
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L6:8 For x > 0, Now we introduce a scaling parameter as x = η ʹ x 2 2 m d 2 ψ ( ʹ x ) dx 2 + eF s x x 0 ( ) ( ʹ x ) = 0 2 2 m 1 2 d 2 ( ʹ x ) d ʹ x 2 + eF s ʹ x ʹ x 0 ( ) ( ʹ x ) = 0 3 2 meF s 2 = 1 ʹ x 0 = E eF s
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L6:9 We desire this to be = 1, so that
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L6:10 This is the “well-known” Airy equation, and the main solutions are the Airy functions Ai( ξ ): What are we trying to say here?
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EEE 434 Quantum Mechanics http://www.eas.asu.edu/~ferry/EEE434.htm L6:11 eV = eF s x We know that the wave function for the lowest bound state must have a single peak, and that it must decay to zero for large x .
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Lecture 06 - David K. Ferry Regents Professor Arizona State...

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