1. Tunneling Outline - Review: Barrier Reflection - Barrier Penetration (Tunneling) - Flash Memory

2. A Simple Potential Step Region 1Region 2 CASE I : E o > V In Region 1: In Region 2:

3. A Simple Potential Step CASE I : E o > V is continuous: Region 1Region 2

4. A Simple Potential Step CASE I : E o > V Region 1Region 2

5. Example from: http://phet.colorado.edu/en/get-phet/one-at-a-timehttp://phet.colorado.edu/en/get-phet/one-at-a-time

6. Quantum Electron Currents Given an electron of mass that is located in space with charge density and moving with momentum corresponding to … then the current density for a single electron is given by

7. A Simple Potential Step CASE I : E o > V Region 1Region 2

8. A Simple Potential Step CASE I : E o > V 1 1 Region 1Region 2

9. IBM Almaden STM of Copper Image originally created by the IBM Corporation. © IBM Corporation. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse.http://ocw.mit.edu/fairuse

10. IBM Almaden Image originally created by the IBM Corporation. © IBM Corporation. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse.http://ocw.mit.edu/fairuse

11. IBM Almaden Image originally created by the IBM Corporation. © IBM Corporation. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse.http://ocw.mit.edu/fairuse

12. A Simple Potential Step CASE II : E o < V In Region 1: In Region 2: Region 1Region 2

13. A Simple Potential Step is continuous: CASE II : E o < V Region 1Region 2

14. A Simple Potential Step CASE II : E o < V Total reflection  Transmission must be zero Region 1Region 2

15. 2a = L Quantum Tunneling Through a Thin Potential Barrier Total Reflection at Boundary Frustrated Total Reflection (Tunneling)

16. CASE II : E o < V Region 1 Region 2Region 3 In Regions 1 and 3: In Region 2: A Rectangular Potential Step for E o < V :

17. A Rectangular Potential Step x=0x=L 2a = L Tunneling Applet: http://www.colorado.edu/physics/phet/dev/quantum-tunneling/1.07.00/http://www.colorado.edu/physics/phet/dev/quantum-tunneling/1.07.00/ Real part of Ψ for E o < V, shows hyperbolic (exponential) decay in the barrier domain and decrease in amplitude of the transmitted wave. Transmission Coefficient versus E o /V for barrier with for E o < V :

18. Electrons tunnel preferentially when a voltage is applied Flash Memory Erased “1” Stored Electrons Programmed “0” Insulating Dielectric SOURCE FLOATING GATE CONTROL GATE CHANNEL Tunnel Oxide Substrate Channel Floating Gate DRAIN Image is in the public domain

19. MOSFET: Transistor in a Nutshell Tunneling causes thin insulating layers to become leaky ! Conducting Channel Image is in the public domain Conduction electron flow Control Gate Semiconductor Image courtesy of J. Hoyt Group, EECS, MIT. Photo by L. Gomez Image courtesy of J. Hoyt Group, EECS, MIT. Photo by L. Gomez

20. UNPROGRAMMEDPROGRAMMED To obtain the same channel charge, the programmed gate needs a higher control-gate voltage than the unprogrammed gate Reading Flash Memory How do we WRITE Flash Memory ? CONTROL GATE FLOATING GATE SILICON CONTROL GATE FLOATING GATE

21. 0L V0V0 x EoEo metal air gap Question: What will T be if we double the width of the gap? Example: Barrier Tunneling Let’s consider a tunneling problem: An electron with a total energy of E o = 6 eV approaches a potential barrier with a height of V 0 = 12 eV. If the width of the barrier is L = 0.18 nm, what is the probability that the electron will tunnel through the barrier?

22. Consider a particle tunneling through a barrier: 1. Which of the following will increase the likelihood of tunneling? a. decrease the height of the barrier b. decrease the width of the barrier c. decrease the mass of the particle 2. What is the energy of the particles that have successfully “escaped”? a. < initial energy b. = initial energy c. > initial energy Multiple Choice Questions 0L V x EoEo Although the amplitude of the wave is smaller after the barrier, no energy is lost in the tunneling process

23. Application of Tunneling: Scanning Tunneling Microscopy (STM) Due to the quantum effect of “barrier penetration,” the electron density of a material extends beyond its surface: material STM tip ~ 1 nm One can exploit this to measure the electron density on a material’s surface: Sodium atoms on metal: STM images Single walled carbon nanotube: VE0E0 STM tip material Image originally created by IBM Corporation Image is in the public domain © IBM Corporation. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse.http://ocw.mit.edu/fairuse

24. Reflection of EM Waves and QM Waves = probability of a particular photon being reflected = probability of a particular electron being reflected Then for optical material when μ=μ 0 :

25. MIT OpenCourseWare http://ocw.mit.edu 6.007 Electromagnetic Energy: From Motors to Lasers Spring 2011 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.http://ocw.mit.edu/terms