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Unformatted text preview: EE372 Assignment 3 Solutions 1. Tunneling a) A tunnel barrier is 5 eV high. What should the width be so that the transmission probability is 10 10 for electrons with 2 eV of energy? b) A tunnel barrier is 5 eV high and 1.2 nm wide. If 10 16 electrons per second attempt to cross the barrier and each electron has an energy of 2 eV, what current flows through the barrier? Answer : a) The probability to tunnel through a barrier is approximately T = 16 E ( V E ) V 2 e 2 αa for aα > 1 For the values given α = q 2 m ( V E ) ¯ h = 8 . 86 × 10 9 m 1 So a = 1 2 α ln V 2 16 E ( V E ) T ! = 1 . 37 × 10 9 m Notice that αa = 12 . 2 which satisfies the condition for the approximation. b) For the values given α = q 2 m ( V E ) ¯ h = 8 . 86 × 10 9 m 1 So the tunneling probability is T = 16 E ( V E ) V 2 e 2 αa = 2 . 2 × 10 9 If 10 16 electrons per second attempt to cross the barrier then the current is I = (10 16 s 1 )(2 . 2 × 10 9 ) e = 3 . 6 × 10 12 A 2. Schr¨ odinger’s equation a) Write down the formal solutions to the onedimensional, timeindependent Schr¨ odinger equation for a constant potential V ( x ) = V . Note that the solutions are different depending on whether E > V or E < V . Do not worry about normalization. b) For the solutions in part (a) in the case E > V , how does the wavelength of the wave function depend on E ? If E increases does the wavelength increase or decrease?...
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 Fall '10
 Johanson/Kasap
 Boundary value problem, Reflection coefficient

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