1. Department of Electronics Nanoelectronics 07 Atsufumi Hirohata 10:00 Tuesday, 27/January/2015 (B/B 103)

2. Quick Review over the Last Lecture Schrödinger equation : ( operator ) ( de Broglie wave ) ( observed results ) For example, ( Eigen value ) ( Eigen function )  H : ( Hermite operator ) Ground state still holds a minimum energy :  ( Zero-point motion )

3. Contents of Nanoelectonics I. Introduction to Nanoelectronics (01) 01 Micro- or nano-electronics ? II. Electromagnetism (02 & 03) 02 Maxwell equations 03 Scholar and vector potentials III. Basics of quantum mechanics (04 ~ 06) 04 History of quantum mechanics 1 05 History of quantum mechanics 2 06 Schrödinger equation IV. Applications of quantum mechanics (07, 10, 11, 13 & 14) 07 Quantum well V. Nanodevices (08, 09, 12, 15 ~ 18)

4. 05 Quantum Well 1D quantum well Quantum tunnelling

5. Classical Dynamics / Quantum Mechanics Major parameters : Quantum mechanicsClassical dynamics Schrödinger equationEquation of motion  : wave function A : amplitude || 2 : probability A 2 : energy

6. 1D Quantum Well Potential A de Broglie wave (particle with mass m 0 ) confined in a square well : General answers for the corresponding regions are x 0 a V0V0 m0m0 -a-a Since the particle is confined in the well, E For E < V 0, CD

7. 1D Quantum Well Potential (Cont'd) Boundary conditions : At x = - a, to satisfy  1 =  2,  1 ’ =  2 ’, At x = a, to satisfy  2 =  3,  2 ’ =  3 ’, For A  0, D - C  0 : For B  0, D + C  0 : For both A  0 and B  0 :   : imaginary number Therefore, either A  0 or B  0.

8. 1D Quantum Well Potential (Cont'd) (i) For A = 0 and B  0, C = D and hence, (ii) For A  0 and B = 0, C = - D and hence, Here, Therefore, the answers for  and  are crossings of the Eqs. (1) / (2) and (3). (1) (2) (3) /2 0  3/2 225/2   Energy eigenvalues are also obtained as  Discrete states * C. Kittel, Introduction to Solid State Physics (John Wiley & Sons, New York, 1986).

9. Quantum Tunnelling In classical theory, Particle with smaller energy than the potential barrier In quantum mechanics, such a particle have probability to tunnel. cannot pass through the barrier. E x 0 a V0V0 E m0m0 For a particle with energy E ( < V 0 ) and mass m 0, Schrödinger equations are Substituting general answers C1C1 A1A1 A2A2

10. Quantum Tunnelling (Cont'd) Now, boundary conditions are Now, transmittance T and reflectance R are  T  0 (tunneling occurs) ! T + R = 1 !

11. Quantum Tunnelling (Cont'd) For  T exponentially decrease with increasing a and ( V 0 - E ) x 0 a V0V0 E m0m0 For V 0 < E, as k 2 becomes an imaginary number, k 2 should be substituted with  R  0 !

12. Quantum Tunnelling - Animation Animation of quantum tunnelling through a potential barrier jtjt jiji jrjr x 0 a * http://www.wikipedia.org/

13. Absorption Coefficient Absorption fraction A is defined as Here, j r = Rj i, and therefore ( 1 - R ) j i is injected. Assuming j at x becomes j - dj at x + dx, jtjt jiji jrjr ( : absorption coefficient) With the boundary condition : at x = 0, j = (1 - R) j i, x 0 a With the boundary condition : x = a, j = (1 - R) j i e -  a, part of which is reflected ; R (1 - R) j i e -  a and the rest is transmitted ; j t = [1 - R - R (1 - R)] j i e -  a