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