Department of Electronics Nanoelectronics 11 Atsufumi Hirohata 12:00 Wednesday, 18/February/2015 (P/L 006)
Quick Review over the Last Lecture Harmonic oscillator : ( Forbidden band ) ( Allowed band ) E k 0 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 : 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
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),...
Orbital Moments * S. Chikazumi, Physics of Ferromagnetism (Oxford University Press, Oxford, 1997). Orbital motion of electron : generates magnetic moment B : Bohr magneton (1.165 Wbm)
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 l E = h H = 0H = 0 H 0H 0 Spin momentum : g = 1 ( J : orbital), 2 ( J : spin) z S
Magnetic Moment
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
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,
Paramagnetism (Cont'd) Now, Using
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
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
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
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 ), * S. Chikazumi, Physics of Ferromagnetism (Oxford University Press, Oxford, 1997). A-site B-site Therefore, total magnetisation is Néel temperature ( T N )