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)

05 Quantum Well

• 1D quantum well

• Quantum tunnelling

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