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Department of Electronics

Nanoelectronics 07. Atsufumi Hirohata. Department of Electronics. 12:00 31/January/2014 Friday (D/L 002). Quick Review over the Last Lecture. Schrödinger equation :. ( de Broglie wave ). ( observed results ). ( operator ). For example,. ( Eigen value ). ( Eigen function ).

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Department of Electronics

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  1. Nanoelectronics 07 Atsufumi Hirohata Department of Electronics 12:00 31/January/2014 Friday (D/L 002)

  2. Quick Review over the Last Lecture Schrödinger equation : ( de Broglie wave ) ( observed results ) ( operator ) 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 :

  6. 1D Quantum Well Potential V0 C D m0 x -a 0 a A de Broglie wave (particle with mass m0) confined in a square well : E General answers for the corresponding regions are Since the particle is confined in the well, For E<V0,

  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 :   : imaginary number For both A0 and B0 : Therefore, either A0 or B0.

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

  9. Quantum Tunnelling V0 C1 A1 E m0 A2 x 0 a In classical theory, E Particle with smaller energy than the potential barrier cannot pass through the barrier. In quantum mechanics, such a particle have probability to tunnel. For a particle with energy E (<V0) and mass m0, Schrödinger equations are Substituting general answers

  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) E m0 V0 x 0 a For  Texponentially decrease with increasing a and (V0-E) ForV0<E, as k2 becomes an imaginary number, k2 should be substituted with  R0 !

  12. Quantum Tunnelling - Animation x 0 a jt ji jr Animation of quantum tunnelling through a potential barrier * http://www.wikipedia.org/

  13. Absorption Coefficient x 0 a jt ji jr Absorption fraction A is defined as Here, jr=Rji, and therefore (1-R) ji is injected. Assuming j at x becomes j-dj at x+dx, ( : absorption coefficient) With the boundary condition : at x=0, j=(1-R)ji, With the boundary condition : x=a, j=(1-R)jie-a, part of which is reflected ; R(1-R)jie-a and the rest is transmitted ; jt=[1-R-R(1-R)]jie-a

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