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

2. 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 )

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  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)
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).

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
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).

9. Magnetic Moment

10. 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).

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).