Department of Electronic Engineering

Nanoelectronics15

Atsufumi Hirohata

09:00 (online) & 12:00 (SLB 118 & online) Monday, 8/March/2021

Quick Review over the Last Lecture 1

Fermi-Dirac distribution Bose-Einstein distribution

Function

Energy dependence

Quantum particlesFermions

e.g., electrons, neutrons, protons and quarks

Bosonse.g., photons, Cooper pairs

and cold Rb

Spins 1 / 2 integer

Properties

At temperature of 0 K, each energy level is occupied by two Fermi particles with opposite spins.® Pauli exclusion principle

At very low temperature, large numbers of Bosons fall into the

lowest energy state.® Bose-Einstein condensation

Contents of NanoelectonicsI. Introduction to Nanoelectronics (01)

01 Micro- or nano-electronics ?

II. Electromagnetism (02 & 03)02 Maxwell equations03 Scholar and vector potentials

III. Basics of quantum mechanics (04 ~ 06)04 History of quantum mechanics 105 History of quantum mechanics 206 Schrödinger equation

IV. Applications of quantum mechanics (07, 10, 11, 13 & 14)07 Quantum well10 Harmonic oscillator11 Magnetic spin13 Quantum statistics 114 Quantum statistics 2

V. Nanodevices (08, 09, 12, 15 ~ 18)08 Tunnelling nanodevices09 Nanomeasurements12 Spintronic nanodevices15 Low-dimensional nanodevices

15 Low-Dimensional Nanodevices

• Quantum wells

• Superlattices

• 2-dimensional electron gas

• Quantum nano-wires

• Quantum dots

Schrödinger Equation in a 3D CubeIn a 3D cubic system, Schrödinger equation for an inside particle is written as :

€

2

2m∂ 2

∂x 2+∂ 2

∂y 2+∂ 2

∂z 2#

$ %

&

' ( ψ x, y,z( ) + E −V( )ψ x, y,z( ) = 0

where a potential is defined as

€

V =0 inside the box : 0 ≤ x ≤ L,0 ≤ y ≤ L,0 ≤ z ≤ L( )+∞ outside the box : x, y,z < 0, x, y,z > L( )

$ % &

' & L

L

Lx

y

z

By considering the space symmetry, the wavefunction is

€

ψ r , t( ) =ψ x, t( )ψ y, t( )ψ z, t( )Inside the box, the Schrödinger equation is rewritten as

In order to satisfy this equation, 1D equations need to be solved :

−ℏ!

2𝑚𝜕!

𝜕𝑥!+𝜕!

𝜕𝑦!+𝜕!

𝜕𝑧!𝜓 𝑥, 𝑡 𝜓 𝑦, 𝑡 𝜓 𝑧, 𝑡 = 𝐸𝜓 𝑥, 𝑡 𝜓 𝑦, 𝑡 𝜓 𝑧, 𝑡

−ℏ!

2𝑚1

𝜓 𝑥, 𝑡𝜕!

𝜕𝑥!𝜓 𝑥, 𝑡 +1

𝜓 𝑦, 𝑡𝜕!

𝜕𝑦!𝜓 𝑦, 𝑡 +1

𝜓 𝑧, 𝑡𝜕!

𝜕𝑧!𝜓 𝑧, 𝑡 = 𝐸" +𝐸# +𝐸$

−ℏ!

2𝑚𝜕!

𝜕𝑥!𝜓 𝑥, 𝑡 = 𝐸"𝜓 𝑥, 𝑡

Schrödinger Equation in a 3D Cube (Cont'd)

To solve this 1D Schrödinger equation, − ℏ!

!$%!

%"!𝜓 𝑥, 𝑡 = 𝐸"𝜓 𝑥, 𝑡 ,

we assume the following wavefunction, 𝜓 𝑥, 𝑡 = 𝜙 𝑥 exp −𝑖𝜔𝑡 , and substitute into the above equation :

By dividing both sides with exp −𝑖𝜔𝑡 ,

For 0 ≤ 𝑥 ≤ 𝐿 , a general solution can be defined as

®

At x = 0, the probability of the particle is 0, resulting 𝜙 0 = 0 : B = 0.

Similarly, 𝜙 𝐿 = 0 : 𝑘" =&"'(

𝑛" = 1, 2, 3,⋯

By substituting into the steady-state Schrödinger equation,

−ℏ!

2𝑚𝜕!

𝜕𝑥!𝜙 𝑥 exp −𝑖𝜔𝑡 = 𝐸"𝜙 𝑥 exp −𝑖𝜔𝑡

−ℏ!

2𝑚𝜕!

𝜕𝑥!𝜙 𝑥 = 𝐸"𝜙 𝑥

𝜙 𝑥 = 𝐴sin 𝑘"𝑥 + 𝐵cos 𝑘"𝑥

𝑘" = ±2𝑚𝐸"ℏ

Schrödinger Equation in a 3D Cube (Cont'd)

By normalising 𝜙 𝑥 = 𝐴sin &"'(𝑥 ,

By considering the symmetry, a particle in a 3D cubic system can be described with the wavefunction :

∴ 𝐴 =2𝐿

Hence, the wave function on x is obtained as

𝜙 𝑥 =2𝐿sin 𝑘"𝑥 𝑘" =

2𝑚𝐸"ℏ

𝜓 𝑥, 𝑦, 𝑧 = 𝜓 𝑥, 𝑡 𝜓 𝑦, 𝑡 𝜓 𝑧, 𝑡 =2𝐿

)!sin 𝑘"𝑥 F sin 𝑘*𝑥 F sin 𝑘+𝑥

Here, the energy eigen values are

𝐸 = 𝐸" + 𝐸* + 𝐸+ =ℏ!

2𝑚 𝑘"! + 𝑘*

! + 𝑘+! =

ℏ!

2𝑚𝜋𝐿

!𝑛"! + 𝑛*! + 𝑛+!

(𝑛", 𝑛*, 𝑛+ = 1, 2, 3,⋯ )

Density of States in Low-Dimensions

𝑛"! + 𝑛*! + 𝑛+! =2𝑚ℏ!

𝐿𝜋

!𝐸

The available states can be obtained as

The available states can be approximated by the volume of the Fermi sphere ( x, y, z ≥ 0 ) :

𝑁! =184𝜋3

2𝑚ℏ!

𝐿𝜋

!𝐸

)

=𝐿)

6𝜋!2𝑚ℏ!

,) !𝐸 ,) ! ny

nz

nxx

y

z

nxi nxi+1

nyinyi+1 nzi

nzi+1

By dividing both sides by L3 :

𝑁 𝐸 =16𝜋!

2𝑚ℏ!

,) !𝐸 ,) ! = N

-

.𝐷 𝐸 𝑑𝐸

The density of states can be obtained as :

𝐷 𝐸 =𝑑𝑁𝑑𝐸 =

23𝜋!

2𝑚ℏ!

,) !𝐸 ,/ !

Density of States in Low-Dimensions

€

D E( )3D ∝ E12

€

D E( )2D ∝ const.

€

D E( )1D ∝ E− 12

D (E) 3D

EF

E

D (E) 2D

EF

E

D (E) 1D

EF

E

® Anderson localisation

Dimensions of NanodevicesBy reducing the size of devices,

3D : bulk

* http://www.kawasaki.imr.tohoku.ac.jp/intro/themes/nano.html

2D : quantum well, superlattice, 2D electron gas

1D : quantum nano-wire

0D : quantum dot

Quantum Well

* http://www.intenseco.com/technology/

A layer (thickness < de Broglie wave length) is sandwiched by high potentials :

Superlattice

* http://www.wikipedia.org/

By employing ultrahigh vacuum molecular-beam epitaxy and sputtering growth,periodically alternated layers of several substances can be fabricated :

Quantum Hall EffectFirst experimental observation was performed by Klaus von Klitzing in 1980 :

* http://www.wikipedia.org/

This was theoretically proposed by Tsuneya Ando in 1974 :

*** http://www.stat.phys.titech.ac.jp/ando/** http://qhe.iis.u-tokyo.ac.jp/research7.html

By increasing a magnetic field, the Fermi level in 2DEG changes.

® Observation of the Landau levels.

† http://www.warwick.ac.uk/~phsbm/qhe.htm

Landau LevelsLev D. Landau proposed quantised levels under a strong magnetic field :

* http://www.wikipedia.org/

B = 0

Ener

gy E

B ¹ 0

hwc

Density of states

Landau levels

Non-ideal Landau levels

EF

increasing B

Fractional Quantum Hall EffectIn 1982, the fractional quantum Hall effect was discovered by Robert B. Laughlin, Horst L. Strömer and Daniel C. Tsui :

* http://nobelprize.org/** H. L. Strömer et al., Rev. Mod. Phys. 71, S298 (1999).

Coulomb interaction between electrons > Quantised potential

¬ Improved quality of samples (less impurity)

Quantum Nano-WireBy further reducing another dimension, ballistic transport can be achieved :

* D. D. D. Ma et al., Science 299, 1874 (2003).

Diffusive / Ballistic TransportIn a macroscopic system, electrons are scattered :

When the transport distance is smaller than the mean free path,

L

l

System length (L) > Mean free path (l)

L

l

® Diffusive transport

System length (L) £ Mean free path (l)

® Ballistic transport

Resistance-Measurement ConfigurationsIn a nano-wire device, 4 major configurations are used to measure resistance :

Hall resistance : Bend resistance :

Longitudinal resistance : Transmission resistance :

iV

i

V

V

i

V

i

local 4-terminal method non-local 4-terminal method

Quantum DotBy further reduction in a dimension, well-controlled electron transport is achieved :

* http://www.fkf.mpg.de/metzner/research/qdot/qdot.html

Nanofabrication - Photolithography

Photolithography :

* http://www.wikipedia.org/

Typical wavelength :Hg lamp (g-line: 436 nm and i-line: 365 nm), KrF laser (248 nm) and ArF laser (193 nm)

** http://www.hitequest.com/Kiss/VLSI.htm

Resolution : > 50 nm (KrF / ArF laser)

Photo-resists :negative-type (SAL-601, AZ-PN-100, etc.)positive-type (PMMA, ZEP-520, MMA, etc.)

*** http://www.ece.gatech.edu/research/labs/vc/theory/photolith.html

Typical procedures :

Nanofabrication - Electron-Beam Lithography

Electron-beam lithography :

* http://www.shef.ac.uk/eee/research/ebl/principles.html

Resolution : < 10 nm

Typical procedures :

NanofabricationNano-imprint :

** K. Goser, P. Glosekotter and J. Diestuhl, Nanoelectronics and Nanosystems (Springer, Berlin, 2004).* http://www.leb.eei.uni-erlangen.de/reinraumlabor/ausstattung/strukturierung.php?lang=en

Current Process TechnologyFront / back end of line :

* http://www.wikipedia.org/

Next-Generation NanofabricationDevelopment of nanofabrication techniques :

* K. Goser, P. Glosekotter and J. Diestuhl, Nanoelectronics and Nanosystems (Springer, Berlin, 2004).

Bottom Up TechnologySelf-organisation (self-assembly) without any control :

* http://www.omicron.de/

Pt nano-wires

** http://www.nanomikado.de/projects.html

Polymer