Department of Electronics Nanoelectronics 07 Atsufumi Hirohata 10:00 Tuesday, 27/January/2015 (B/B 103
Dec 31, 2015
Department of Electronics
Nanoelectronics
07
Atsufumi Hirohata
10:00 Tuesday, 27/January/2015 (B/B 103)
Quick Review over the Last Lecture
Schrödinger equation :
( operator )
( de Broglie wave )
( observed results )
For example,
( Eigen value )
( Eigen function )
H : ( Hermite operator )
Ground state still holds a minimum energy :
( Zero-point motion )
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)
Classical Dynamics / Quantum Mechanics
Major parameters :
Quantum mechanics Classical dynamics
Schrödinger equation Equation of motion
: wave function A : amplitude
||2 : probability A2 : energy
1D Quantum Well Potential
A de Broglie wave (particle with mass m0) confined in a square well :
General answers for the corresponding regions are
x0 a
V0
m0
-a
Since the particle is confined in the well,
E
For E < V0,
C D
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.
1D Quantum Well Potential (Cont'd)
(i) For A = 0 and B 0, C = D and hence,
(ii) For A 0 and B = 0, C = - D and hence,
Here,
Therefore, the answers for and are crossings of the Eqs. (1) / (2) and (3).
(1)
(2)
(3)/20 3/2 2 5/2
Energy eigenvalues are also obtained as
Discrete states
* C. Kittel, Introduction to Solid State Physics (John Wiley & Sons, New York, 1986).
Quantum Tunnelling
In classical theory,
Particle with smaller energy than the potential barrier
In quantum mechanics, such a particle have probability to tunnel.
cannot pass through the barrier.
E
x0 a
V0
Em0
For a particle with energy E (< V0) and mass m0,
Schrödinger equations are
Substituting general answers
C1A1
A2
Quantum Tunnelling (Cont'd)
Now, boundary conditions are
Now, transmittance T and reflectance R are
T 0 (tunneling occurs) ! T + R = 1 !
Quantum Tunnelling (Cont'd)
For
T exponentially decrease
with increasing a and (V0 - E)
x0 a
V0
Em0
For V0 < E, as k2 becomes an imaginary number,
k2 should be substituted with
R 0 !
Quantum Tunnelling - Animation
Animation of quantum tunnelling through a potential barrier
jtji
jrx
0 a
* http://www.wikipedia.org/
Absorption Coefficient
Absorption fraction A is defined as
Here, jr = Rji, and therefore (1 - R) ji is injected.
Assuming j at x becomes j - dj at x + dx,
jtji
jr( : absorption coefficient)
With the boundary condition : at x = 0, j = (1 - R) ji,
x0 a
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