Department of Electronics

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).