Department of Electronics

Introductory Nanotechnology

~ Basic Condensed Matter Physics ~

Atsufumi Hirohata

Quick Review over the Last Lecture 1

E

(Conduction) band

( Valence ) band

( Band gap ) (conduction electron)

( positive hole )

Shockley model :

Semiconductors :

Elemental Compound

Si, Ge, … GaAs, InAs, ...

Intrinsic Extrinsic

• EF = (EC - EV) / 2 • np = const.

• doping • n-type : donor • p-type : acceptor

• np = const.

EC (Eg)

EF

EA

EV (0)

EC (Eg)

ED

EF

EV (0)

Quick Review over the Last Lecture 2

Temperature dependence of an extrinsic semiconductor :

Schottky barrier height

EFM EFS

M n-S

EV

ED EC M - S

- - - -

+ + + +

depletion layer

Metal - semiconductor junction :

J

V

JFoward = J0 expqV

kBT

JReverse = J0

pn junction :

(built-in potential)

Contents of Introductory Nanotechnology

First half of the course :

Basic condensed matter physics

1. Why solids are solid ?

2. What is the most common atom on the earth ?

3. How does an electron travel in a material ?

4. How does lattices vibrate thermally ?

5. What is a semi-conductor ?

6. How does an electron tunnel through a barrier ?

Second half of the course :

Introduction to nanotechnology (nano-fabrication / application)

7. Why does a magnet attract / retract ?

8. What happens at interfaces ?

How Does an Electron

Tunnel through a Barrier ?

• De Broglie wave

• Schrödinger equation

• 1D quantum well

• Quantum tunneling

• Reflectance / transmittance

• Optical absorption

• Direct / indirect band gap

Electron Interference

Davisson-Germer experiment in 1927 :

Electrons are introduced to a screen through two slits.

Electron as a particle

should not interfere.

Photon (light) as a wave

* http://www.wikipedia.org/

-

+

Electron interference observed !

Wave-particle duality

Scrödinger's Cat

Thought experiment proposed by E. Schrödinger in 1935

Radioactive substance

Hydrocyanic acid

• if radioactive .

• if and has been .

• if

* http://www.wikipedia.org/

The quantum :

superposition

The superposition is lost :

• the observer s box and .

• then,

De Broglie Wave

Wave packet :

contains number of waves, of which

amplitude describes probability of the

presence of a particle.

=h

m0v

where : wave length, h : Planck constant

and m0 : mass of the particle.

* http://www.wikipedia.org/

de Broglie hypothesis

(1924 PhD thesis 1929 Nobel prize)

According to the mass-energy equivalence :

E = m0c2= m0c c = p

where p : momentum and : frequency.

By using E = h ,

=h

p=

h

m0v

Schrödinger Equation

In order to express the de Broglie wave, Schrödinger equation is introduced in 1926 :

h2

2m2 + E V( ) = 0 2 =

2

x 2+

2

y 2+

2

z 2

E : energy eigenvalue and : wave function

Wave function represents probability of the presence of a particle 2=

*

* : complex conjugate (e.g., z = x + iy and z* = x - iy)

Propagation of the probability (flow of wave packet) :

j =h

2mi* *( )

Operation = observation :

h2

2m2 = E V( )

operator

de Broglie wave

observed results

1D Quantum Well Potential

A de Broglie wave (particle with mass m0) confined in a square well :

h2

2m0

d 21

dx 2+ E V0( ) 1 = 0 x < a( )

h2

2m0

d 22

dx 2+ E 2 = 0 a < x < a( )

h2

2m0

d 23

dx 2+ E V0( ) 3 = 0 a < x( )

General answers for the corresponding regions are

1 = Ce x +C1ex x < a( )

2 = A sin x + B cos x a < x < a( )

3 = De x +D1ex a < x( )

x 0 a

V0

m0

-a

=2m0E

h

=2m0 V0 E( )

h

Since the particle is confined in the well,

E

1, 3 0 x ±( )

For E < V0, C1 = 0, D1 = 0

C D

1D Quantum Well Potential (Cont'd)

Boundary conditions :

At x = -a, to satisfy 1 = 2, A sin a + B cos a = Ce a

1’ = 2’, A cos a + B sin a = Ce a

At x = a, to satisfy 2 = 3, A sin a + B cos a = De a

2’ = 3’, A cos a B sin a = De a

2A sin a = D C( )e a

2 A cos a = D C( )e a

2B cos a = C +D( )e a

2 B sin a = C +D( )e a

For A 0, D - C 0 : cot a =

For B 0, D + C 0 : tan a =

For both A 0 and B 0 : tan2 a = 1 : 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,

tan = a = , a =( )

(ii) For A 0 and B = 0, C = -D and hence,

cot =

Here,

=2m0E

h, =

2m0 V0 E( )h

2+

2=2m0V0

h2

2+

2=2m0V0a

2

h2

Therefore, the answers for and are crossings of the Eqs. (1) / (2) and (3).

(1)

(2)

(3) /2 0 3 /2 2 5 /2

Energy eigenvalues are also obtained as

E =h2

2m0a2

2

Discrete states

Quantum Tunneling

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

x 0 a

V0

E m0

For a particle with energy E (< V0) and mass m0,

Schrödinger equations are

h2

2m0

d 2

dx 2+ E = 0 x < 0, a < x( )

h2

2m0

d 2

dx 2+ E V0( ) = 0 0 < a < x( )

Substituting general answers k1 = 2m0E h , k2 = 2m0 V0 E( ) h

= A1 exp ik1x( ) + A2 exp ik1x( ) x < 0( )

= B1 exp k2x( ) + B2 exp k2x( ) 0 < a < x( )

= C1 exp ik1x( ) +C2 exp ik1x( ) a < x( )

C1 A1

A2

Quantum Tunneling (Cont'd)

Now, boundary conditions are

A1 + A2 = B1 + B2, ik1 A1 A2( ) = k2 B1 B2( ) x = 0( )

B1 exp k2a( ) + B2 exp k2a( ) = C1 exp ik1a( ), k2 B1 exp k2a( ) B2 exp k2a( )[ ] = ik1C1 exp ik1a( ) x = a( )

C1A1

=4ik1k2 exp ik1a( )

k2 + ik1( )2exp k2a( ) k2 ik1( )

2exp k2a( )

A2A1

=k12

+ k22( ) exp k2a( ) exp k2a( )[ ]

k2 ik1( )2exp k2a( ) k2 + ik1( )

2exp k2a( )

Now, transmittance T and reflectance R are

T =C1A1

2

=4k1

2k22

k12

+ k22( )2sinh2 k2a( ) + 4k1

2k22

=4E V0 E( )

V02 sinh2 a 2b( ) + 4E V0 E( )

R =A2A1

2

=k12

+ k22( )2sinh2 k2a( )

k12

+ k22( )2sinh2 k2a( ) + 4k1

2k22

=V0 sinh

2 a 2b( )V02 sinh2 a 2b( ) + 4E V0 E( )

b =h

2 2m0 V0 E( )

T 0 (tunneling occurs) !

Quantum Tunneling (Cont'd)

For

T exponentially decrease

with increasing a and (V0 - E)

V0 E >> h

2 2m0a2

h a << 2m0 V0 E( ) , a 2b >> 1

V02 sinh2 a 2b( ) + 4E V0 E( ) V0

2 sinh2 a 2b( ) V02 exp a b( ) V0

2 exp2 2m0 V0 E( )a

h

T4E V0 E( )

V02

exp2 2m0 V0 E( )a

h

x 0 a

V0

E m0

For V0 < E, as k2 becomes an imaginary number,

k2 should be substituted with

k2 =2m0 E V0( )

h k2 ik2

T =4k1

2k22

k12 k2

2

2

sin2 k2 a

+ 4k1

2k22

=4E E V0( )

V02 sin2 k2 a

+ 4E E V0( )

R =

k12

+ k22

2

sin2 k2 a

k12

+ k22

2

sin2 k2 a

+ 4k1

2k22

=V0 sin

2 k2 a

V02 sin2 k2 a

+ 4E E V0( )

R 0 !

Reflectance and Transmittance

At an energy level E, wave function is expressed as = x( ) exp iEt h( )

2= x( )

22

t=

x( )2

t= 0

According to the equation of continuity : t+ div j = 0 div j =

jxx

+jyy

+jzz

div j = 0

For 1D, djxdx

= 0 j = jx = const.

At the incident side, the incident wave i and the reflection wave

r satisfy

= i + r j = ji + jr

Here,

j =h

2m0i* * ( ), ji =

h

2m0ii*

i i i * ( ), jr =

h

2m0ir*

r r r * ( )

At the transmission side, only the transmission wave t exists, and thus j = jt

ji + jr = j t ji = j t + jr

1=jtji+jrj i

1=T + R

T : transmittance and R : reflectance (j

t / j

i) (|j

r| / j

i)

jt

ji

jr

Transistor and Esaki Diode

First bipolar transistor (transfer resistor) was invented

by . , . . in 1947 :

Tunneling diode was invented by L. Esaki in 1958 :

* http://www.wikipedia.org/; http://photos.aip.org/; ** S. M. Sze, Physics of Semiconductor Devices (John Wiley, New York, 1981).

First observation of tunneling effect !

Absorption Coefficient

Absorption fraction A is defined as

A + R +T = 1Here, j

r = Rj

i, and therefore (1 - R) j

i is injected.

Assuming j at x becomes j - dj at x + dx,

dj = jdx

jt

ji

jr

( : absorption coefficient)

With the boundary condition : at x = 0, j = (1 - R) ji,

j = 1 R( ) j i exp x( )

x 0 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)] j

i e - a

jt = 1 R( )2ji exp x( )

T =jtji= 1 R( )

2exp x( )

Optical Absorption

* M. Sakata, Solid State Physics (Baifukan, Tokyo, 1989).

Absorp

tion c

oeff

icie

nt

Wavelength

Conduction band

Valence band

Fundam

enta

l absorp

tion

Fundam

enta

l absorp

tion

edge

Exciton a

bsorp

tion

edge

Exciton absorption

Impurity absorption

Impurity absorption

Trap level

Trap level

Exciton level

Lattice vibration

Conduction absorption

Band gap

Conduction carrier F-c

entr

e

Semiconductor Band Gap

* M. Sakata, Solid State Physics (Baifukan, Tokyo, 1989).

Absorp

tion c

oeff

icie

nt

Conduction band

Valence band

Absorp

tion c

oeff

icie

nt

Valence band

Conduction band

Direct transition starts

Indrect transition starts

Direct transition starts

Excited electrons will be recombine with holes with emitting photon.

Light emitting diode (LED)

Semiconductor Band Gap in Si, Ge and GaAs

* M. P. Marder, Condensed Matter Physics (John-Wiley, New York, 2000).

Photo Diode

Photovoltaic effect :

* M. Sakata, Solid State Physics (Baifukan, Tokyo, 1989).

Photo- current

Photo-voltage

Direct energy conversion

Photo diode

Solar cell