1. Department of Electronics Nanoelectronics 11 Atsufumi Hirohata 12:00 Wednesday, 18/February/2015 (P/L 006)

2. Quick Review over the Last Lecture Harmonic oscillator : ( Forbidden band ) ( Allowed band ) E k 0 1st 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 : z component is calculated to be 0 p r L In order to convert L z into an operator, 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) Thus, the eigenvalue for L 2 is  azimuthal quantum number (defines the magnitude of L ) * S. Chikazumi, Physics of Ferromagnetism (Oxford University Press, Oxford, 1997). Similarly, for L z,  magnetic quantum number (defines the magnitude of L z ) For a simple electron rotation, L LzLz  Orientation of L : quantized In addition, principal quantum number : defines electron shells n = 1 (K), 2 (L), 3 (M),...

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

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

9. Magnetic Moment

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

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

12. Paramagnetism (Cont'd) Now, Using

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

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

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

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 ), * S. Chikazumi, Physics of Ferromagnetism (Oxford University Press, Oxford, 1997). A-site B-site Therefore, total magnetisation is  Néel temperature ( T N )