Nanoelectronics 11 Atsufumi Hirohata Department of Electronic Engineering

Nanoelectronics 11 Atsufumi Hirohata Department of Electronic Engineering 09: 00 (online) & 12: 00 (SLB 118 & online) Monday, 22/February/2021

Quick Review over the Last Lecture Harmonic oscillator : E ( Allowed band ) ( Forbidden band ) ( Allowed band ) k 0 2 nd 1 st 2 nd ( 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 0 r By changing into a polar coordinate system, Similarly, i Therefore, In quantum mechanics, observation of state = R is written as p

Origin of Magnetism (Cont'd) Thus, the eigenvalue for L 2 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 ml Zeeman splitting : For H atom, energy levels are split under H dependent upon ml. 2 1 0 -1 -2 l 2 E = h Spin momentum : 1 0 -1 1 H=0 H 0 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 : Eex : minimum for parallel / antiparallel configurations Si Sj Dipole moment arrangement : Paramagnetism Exchange integral Jex : exchange integral Antiferromagnetism antiferromagnetism Atom separation [Å] 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 + d. U, Boltzmann distribution Sum of the moments along z direction is between -J and +J (MJ : z component of M) Here, H

Paramagnetism (Cont'd) Now, 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 + w. M, and replacing a with x, Hm Spontaneous magnetisation at H = 0 is obtained as Using M 0 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), B Therefore, total magnetisation is A-site B-site Néel temperature (TN) * S. Chikazumi, Physics of Ferromagnetism (Oxford University Press, Oxford, 1997).