Department of Electronics Nanoelectronics 11 Atsufumi Hirohata Department of Electronics 13:00 Monday, 20/February/2017 (P/T 006)

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

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

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

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  r p By changing into a polar coordinate system, Similarly, Therefore, In quantum mechanics, observation of state  = R is written as

Origin of Magnetism (Cont'd) 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, L Lz  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).

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).

Spin Moment and Magnetic Moment Zeeman splitting : ml 2 1 -1 -2 l E = h H = 0 H  0 For H atom, energy levels are split under H dependent upon ml. Spin momentum : z S  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).

Magnetic Moment

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

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,

Paramagnetism (Cont'd) Now, Using Using

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).

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).

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).

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

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).