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Journey Into Nanoelectronics: Theory & Applications of Quantum Mechanics

Explore the fundamentals of nanoelectronics, including electromagnetism, quantum mechanics, applications, nanodevices, and the origins of magnetism. Delve into the spin, orbital moments, exchange energy, and various magnetic interactions in this detailed quick review.

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Journey Into Nanoelectronics: Theory & Applications of Quantum Mechanics

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  1. Nanoelectronics 11 Atsufumi Hirohata Department of Electronics 09:00 19/February/2014 Wednesday (G 013)

  2. Quick Review over the Last Lecture Harmonic oscillator : E ( Allowed band ) ( Forbidden band ) ( Allowed band ) ( Forbidden band ) ( Allowed band ) k 0 1st 2nd 2nd ( Brillouin zone )

  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 10 Harmonic oscillator 11 Magnetic spin V. Nanodevices (08, 09, 12, 15 ~ 18) 08 Tunnelling nanodevices 09 Nanomeasurements

  4. 11 Magnetic spin • Origin of magnetism • Spin / orbital moment • Paramagnetism • Ferromagnetism • Antiferromagnetism

  5. Origin of Magnetism Angular momentum L is defined with using momentum p : L z component is calculated to be In order to convert Lz into an operator, p 0 r p By changing into a polar coordinate system, Similarly, Therefore, In quantum mechanics, observation of state =R is written as

  6. Origin of Magnetism (Cont'd) Lz L Thus, the eigenvalue for L2 is  azimuthal quantum number (defines the magnitude of L) Similarly, for Lz,  magnetic quantum number (defines the magnitude of Lz) For a simple electron rotation,  Orientation of L : quantized In addition, principal quantum number : defines electron shells n = 1 (K), 2 (L), 3 (M), ... * S. Chikazumi, Physics of Ferromagnetism (Oxford University Press, Oxford, 1997).

  7. Orbital Moments Orbital motion of electron : generates magnetic moment  B : Bohr magneton (1.16510-29 Wbm) * S. Chikazumi, Physics of Ferromagnetism (Oxford University Press, Oxford, 1997).

  8. Spin Moment and Magnetic Moment ml 2 l 1 2 0 -1 -2 E = h 1 0 1 -1 z H=0 H0 S Zeeman splitting : For H atom, energy levels are split under H dependent upon ml. Spin momentum :  g=1 (J : orbital), 2 (J : spin) Summation of angular momenta : Russel-Saunders model J=L+S Magnetic moment : * S. Chikazumi, Physics of Ferromagnetism (Oxford University Press, Oxford, 1997).

  9. Magnetic Moment

  10. Exchange Energy and Magnetism ferromagnetism Exchange integral Jex antiferromagnetism Atom separation [Å] Exchange interaction between spins : Sj Si  Eex : minimum for parallel / antiparallel configurations  Jex : exchange integral Dipole moment arrangement : Paramagnetism Antiferromagnetism Ferromagnetism Ferrimagnetism * K. Ota, Fundamental Magnetic Engineering I (Kyoritsu, Tokyo, 1973).

  11. Paramagnetism Applying a magnetic field H, potential energy of a magnetic moment with  is  m rotates to decrease U.  Assuming the numbers of moments with  is n and energy increase with +d is +dU, H  Boltzmann distribution Sum of the moments along z direction is between -J and +J (MJ : z component of M) Here,

  12. Paramagnetism (Cont'd) Now, Using Using

  13. Paramagnetism (Cont'd) Therefore,  BJ(a) : Brillouin function For a   (H   or T  0), Ferromagnetism For J  0, M  0 For J   (classical model),  L(a) : Langevin function * S. Chikazumi, Physics of Ferromagnetism (Oxford University Press, Oxford, 1997).

  14. Ferromagnetism Weiss molecular field : (w : molecular field coefficient, M : magnetisation) In paramagnetism theory, Substituting H with H+wM, and replacing a with x, Hm Spontaneous magnetisation at H=0 is obtained as Using M0 at T=0, For x<<1, Assuming T= satisfies the above equations, (TC) : Curie temperature * H. Ibach and H. Lüth, Solid-State Physics (Springer, Berlin, 2003).

  15. Ferromagnetism (Cont'd) For x<<1, Therefore, susceptibility  is (C : Curie constant)  Curie-Weiss law ** S. Chikazumi, Physics of Ferromagnetism (Oxford University Press, Oxford, 1997).

  16. Spin Density of States * H. Ibach and H. Lüth, Solid-State Physics (Springer, Berlin, 2003).

  17. Antiferromagnetism By applying the Weiss field onto independent A and B sites (for x<<1), A-site B-site Therefore, total magnetisation is  Néel temperature (TN) * S. Chikazumi, Physics of Ferromagnetism (Oxford University Press, Oxford, 1997).

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