EE372_Solutions3

# EE372_Solutions3 - EE372 Assignment 3 Solutions 1 Tunneling...

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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 one-dimensional, time-independent 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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EE372_Solutions3 - EE372 Assignment 3 Solutions 1 Tunneling...

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