1. Department of Electronics
Nanoelectronics 03
Atsufumi Hirohata
Department of Electronics
13:00 Monday, 23/January/ (P/T 006)

2. Quick Review over the Last Lecture
Maxwell equations :
 Time-independent case : ( Ampère’s / Biot-Savart ) law
 ( Faraday’s ) law of induction
( Time-dependent equations )
 ( Gauss ) law
( Initial conditions )
 ( Gauss ) law for magnetism
( Assumptions )
Electromagnetic wave :
( Electric ) field
( Magnetic ) field
Propagation direction
propagation speed :
in a vacuum,

3. Contents of Nanoelectronics
I. Introduction to Nanoelectronics (01)
01 Micro- or nano-electronics ?
II. Electromagnetism (02 & 03)
02 Maxwell equations
03 Scalar and vector potentials
III. Basics of quantum mechanics (04 ~ 06)
IV. Applications of quantum mechanics (07, 10, 11, 13 & 14)
V. Nanodevices (08, 09, 12, 15 ~ 18)

4. 03 Scalar and Vector Potentials
Scalar potential 
Vector potential A
Lorentz transformation

5. Maxwell Equations Maxwell equations :
E : electric field, B : magnetic flux density,
H : magnetic field, D : electric flux density,
J : current density and  : charge density
Supplemental equations for materials :
 Definition of an electric flux density
 Definition of an magnetic flux density
 Ohm’s law

6. Electromagnetic Potentials
Scalar potential  and vector potential A are defined as
(1)
(2)
By substituting these equations into Eq. (2),
(3)
(4)
Here,
 Similarly, y- and z-components become 0.
Also,
 Satisfies Eq. (2).

7. Electromagnetic Potentials (Cont'd)
(1)
(2)
By substituting these equations into Eq. (4),
(3)
(4)
 Satisfies Eq. (4).

8. Electromagnetic Potentials (Cont'd)
By assuming x as a differentiable function, we define
 Gauge transformation
Here, A’ and ’ provide E and B as the same as A and .
Therefore, electromagnetic potentials A and  contains uncertainty of x.
In particular, when A and  satisfies the following condition :
 Laurenz gauge
Under this condition, Eqs. (1) and (3) are expressed as
(1)
(2)
(3)
 Equation solving at home
(4)

9. Einstein's Theory of Relativity
In 1905, Albert Einstein proposed the theory of special relativity :
Lorentz invariance for Maxwell’s equations (1900)
Poincaré proved the Lorentz invariance for dynamics.
 Lorentz invariance in any inertial coordinates
Lorenz gauge
Speed of light (electromagnetic wave) is constant. *

10. Scalar Potential 
Scholar potential holds the following relationship with a force :
The concept was first introduced by Joseph-Louis Lagrange in 1773,
and named as scalar potential by George Green in 1828. * **

11. Scalar Potential  and Vector Potential A
Scalar potential holds the following relationship with a force :
Electrical current
Voltage potential
induced by a positive charge
Vector potential around a current
rot A
Magnetic
field
Vector potential A
(decrease with increasing
the distance from the current)
Electric field E
(induced along the direction to decrease
the voltage potential)
Magnetic field H
(induced along the axis of the rotational flux
of the vector potential, rot A) *

12. Vector Potential A Faraday found electromagnetic induction in 1831 :
Faraday considered that an “electronic state” of the coil
can be modified by moving a magnet.
 induces current flow.
In 1856, Maxwell proposed a theory
with using a vector potential instead of “electronic state.”
 Rotational spatial distribution of A generates a magnetic flux B.
 Time evolution of A generates an electric field E. *

13. Maxwell's Vector Potential
magnetic line of force
vector potential
electrical current
electrical
current
magnetic field
By the rotation of the vector potentials in the opposite directions,
rollers between the vector potentials move towards one direction.
 Ampère’s law
After the observation of a electromagnetic wave,
E and B : physical quantities
A : mathematical variable *

14. Aharonov-Bohm Effect
Yakir Aharonov and David Bohm theoretically predicted in 1959 :
Electron can modify its phase
without any electrical / magnetic fields.
electron source
coil
shield
phase shift
interference
magnetic field * **

15. Observation of the Vector Potential
In 1982, Akira Tonomura proved the Aharonov-Bohm effect : Nb
NiFe
No phase shift
Phase shift =
wavelength / 2 *
** org/jpn/books/kaishikiji/200012/ html