1. Department of Electronics
Nanoelectronics 07
Atsufumi Hirohata
Department of Electronics
13:00 Monday, 4/February/ (P/T 006)

2. Quick Review over the Last Lecture
Schrödinger equation :
( de Broglie wave )
( observed results )
( operator )
For example,
( Eigen value )
( Eigen function )
 H : ( Hermite operator )
Ground state still holds a minimum energy :
 ( Zero-point motion )

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
V. Nanodevices (08, 09, 12, 15 ~ 18)

4. 05 Quantum Well
1D quantum well
Quantum tunnelling

5. Classical Dynamics / Quantum Mechanics
Major parameters :
Quantum mechanics
Classical dynamics
Schrödinger equation
Equation of motion
 : wave function
A : amplitude
||2 : probability
A2 : energy

6. 1D Quantum Well Potential
A de Broglie wave (particle with mass m0) confined in a square well : x a V0 m0 -a E C D
General answers for the corresponding regions are
Since the particle is confined in the well,
For E < V0,

7. 1D Quantum Well Potential (Cont'd)
Boundary conditions :
At x = -a, to satisfy 1 = 2,
1’ = 2’,
At x = a, to satisfy 2 = 3,
2’ = 3’,
For A  0, D - C  0 :
For B  0, D + C  0 :
For both A  0 and B  0 :
  : imaginary number
Therefore, either A  0 or B  0.

8. 1D Quantum Well Potential (Cont'd)
(i) For A = 0 and B  0, C = D and hence,
(1)
(ii) For A  0 and B = 0, C = - D and hence,
(2)
/2 
3/2 2
5/2  
Here,
(3)
Therefore, the answers for  and  are crossings of the Eqs. (1) / (2) and (3).
Energy eigenvalues are also obtained as
 Discrete states
* C. Kittel, Introduction to Solid State Physics (John Wiley & Sons, New York, 1986).

9. Quantum Tunnelling In classical theory, E
Particle with smaller energy than the potential barrier
cannot pass through the barrier.
In quantum mechanics, such a particle have probability to tunnel.
For a particle with energy E (< V0) and mass m0,
Schrödinger equations are x a V0 E m0 C1 A1 A2
Substituting general answers

10. Quantum Tunnelling (Cont'd)
Now, boundary conditions are
Now, transmittance T and reflectance R are
 T  0 (tunneling occurs) !
 T + R = 1 !

11. Quantum Tunnelling (Cont'd)
For
 T exponentially decrease
with increasing a and (V0 - E) x a V0 E m0
For V0 < E, as k2 becomes an imaginary number,
k2 should be substituted with
 R  0 !

12. Quantum Tunnelling - Animation
Animation of quantum tunnelling through a potential barrier jt ji jr x a *

13. Absorption Coefficient
Absorption fraction A is defined as jt ji jr
Here, jr = Rji, and therefore (1 - R) ji is injected.
Assuming j at x becomes j - dj at x + dx,
( : absorption coefficient) x a
With the boundary condition : at x = 0, j = (1 - R) ji,
With the boundary condition : x = a, j = (1 - R) jie -a,
part of which is reflected ; R (1 - R) ji e -a
and the rest is transmitted ; jt = [1 - R - R (1 - R)] ji e -a