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Department of Electronics
Nanoelectronics
05
Atsufumi Hirohata
12:00 Wednesday, 21/January/2015 (P/L 006)
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Quick Review over the Last Lecture
Light quantum :
E = ( h )
( Compton ) scattering
Electron ( interference )
( De Broglie ) wave
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Contents of Nanoelectonics
I. Introduction to Nanoelectronics (01) 01 Micro- or nano-electronics ?
II. Electromagnetism (02 & 03) 02 Maxwell equations 03 Scalar 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)
V. Nanodevices (08, 09, 12, 15 ~ 18)
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05 History of Quantum Mechanics 2
• Rutherford’s model
• Bohr’s model
• Balmer series
• Uncertainty principle
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Early Models of an Atom
In 1904, J. J. Thomson proposed a plum pudding model :
Negatively charged “plums” (electrons) are surrounded by
positively charged “pudding.”
* http://www.wikipedia.org/** http://www.nararika.com/butsuri/kagakushi/genshi/genshiron.htm
In 1904, Hantaro Nagaoka proposed a Saturn model :
Negatively charged electrons rotate around positively-charged core.
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Rutherford's Model
In 1909, Ernest Rutherford carried out a Au foil experiment :
-ray was introduced onto a very thin Au foil.
* http://www.wikipedia.org/
cannot be explained by the plum pudding model,
and the Saturn model was adopted.
Rutherford back scattering was observed.
The size of the core is estimated to be
10 -14 m.
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Rutherford Back Scattering
Powerful tool for materials analysis :
* http://www.toray-research.co.jp/kinougenri/hyoumen/hyo_006.html
Analyser
Sample atoms
Example
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Limitation of Rutherford's Model
In classical electromagnetism,
An electron rotating around the core loses its energy
by irradiating electromagnetic wave,
and falls into the positively-charged core.
In 1913, Niels H. D. Bohr proposed a quantum rule :
An electron can permanently rotate around the core
when occupying an orbital with
me : electron mass, v : electron speed, r : orbital radius,
n : quantum number and h : Planck constant
stable state
energy levels (n = 1, 2, 3, …)
* http://www.wikipedia.org/
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Bohr's Model
Allowed transitions between the energy levels :
From the energy level of En to that of En’,
a photon is absorbed when En’ - En > 0.
a photon is released when En’ - En’ < 0.
* http://www.wikipedia.org/
Meaning of the quantum rule :
De Broglie wave length is defined as
By substituting this relationship into the quantum rule,
Electron as a standing wave in an orbital.
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Proof of Bohr's Model
In 1914, James Franck and Gustav L. Hertz proved discrete energy levels :
An acceleration voltage is tuned to allow
diluted gas to absorb the energy.
* http://nobelprize.org/** http://www.wikipedia.org/
A proof of the quantum rule
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Balmer Series in a Hydrogen Atom
In 1885, Johann Jakob Balmer proposed an empirical formula :
Balmer series observation :
Balmer formula :
* http://www.wikipedia.org/
: wavelength, B : constant (364.56 nm), n = 2 and
m : an integer (m > n)
A proof of the discrete electron orbitals
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Rydberg Formula
In 1888, Johannes R. Rydberg generalised the Balmer formula :
Rydberg formula for Hydrogen :
Rydberg formula for atoms :
* http://www.wikipedia.org/
RH : Rydberg constant (10973731.57 m−1)
vac : wavelength of the light emitted in a vacuum,Z : atomic number, m and n : integers
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Balmer Series in a Hydrogen Atom
Other series were also found :
* http://www.bigs.de/en/shop/htm/termsch01.html
Lyman series(1906)
Ultra violet
Balmer series(1885)
Visible light
Brackett series(1922)
Near infrared
Paschen series(1908)Infrared
Pfund series(1924)
Far infrared
Humphreys series(1953)
Far infrared
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Heisenberg's Thought Experiment
In 1927, Werner Karl Heisenberg proposed the uncertainty principle :
Resolution of a microscope is defined as
* http://www.wikipedia.org/
In order to minimise ,
Larger
Small
Here,
Small large p (damage to samples)
Also, larger difficult to identify paths
incident wave
x-direction electron
In addition, incident wave can be reflected within
Momentum along x is within
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Observation of the Uncertainty Principle
Zero-point motion :
He does not freeze near 0 K under atmospheric pressure.
Spin fluctuation in an itinerant magnet :
* http://www.e-one.uec.ac.jp/~kuroki/cobalt.html
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Precise Quantum Physical Definition
Relationship between an observation error in position and disturbance in momentum :
Heisenberg’s uncertainty principle using operators :
Heisenberg’s uncertainty principle using standard deviations :
Here, communication relation :
If A and B are commutative operators, the right-hand side is 0. This leads the error and disturbance to be 0.
However, A and B are not commutative operators, the right-hand side is not 0. This leads the error and disturbance to have a trade-off.
→ Precise definition
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Corrections to Heisenberg’s Uncertainty Principle
Ozawa’s relationship : *
Optical proof : **
* M. Ozawa, Phys. Rev. A 67, 042105 (2003);** S.-Y. Baek et al., Sci. Rep. 3, 2221 (2013).
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Experimental Data
Error and disturbance measured :
** S.-Y. Baek et al., Sci. Rep. 3, 2221 (2013).
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Ozawa’s Relationship
Ozawa’s relationship :
** S.-Y. Baek et al., Sci. Rep. 3, 2221 (2013).