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Electrical, optical and magnetic properties of materials
7/13/2010 Hand-out #1
Instructor:
Prof. Nate Newman
Lawrence Professor of Solid State Science and
Director of Solid State Science
Arizona State University
Class time:
Class time: 9:00 - 11:00 AM, TWTh
Class Location: Department de Fisica d Qimica,
Universidade Estadual Paualista- UNESP
Useful Textbook: R.E. Hummel, Electronic Properties of Materials,
Springer Verlag; ; 3rd edition (December 2000),
ISBN: 038795144X
--optional reference: N. W. Ashcroft and N. D.Mermin,
Solid State Physics, Holt Rinhart and Winston, NY.
N. Newman, [email protected], Lecture 1, page 1
Course Outline
This course is designed to give the student a basic
understanding of the mechanisms responsible for the
electrical, optical and magnetic properties of solids.
Fundamental models of the electronic and lattice
properties of solids are studied in detail. Comparison to
experiment is used to illustrate the accuracy and
limitations of the models. Applications of each class of
materials are used to highlight the impact of these
methods on modern technology.
N. Newman, [email protected], Lecture 1, page 2
Syllabus
.
1. Electrons in solids: particles or waves?
Wave packets & the Schrodinger equation
2. Energy bands of metals, semiconductors and insulators
Energy bands of real material systems; theory and experiment
3. Electrical properties
Normal conduction in solids
Superconductivity
4. Statistics of quasi-particle
5. Optical Properties
Oscillator model
Kramers-Kronig analysis
Optical constants of real systems
Luminescence
Stimulated emission
6. Magnetic properties
Diamagnetism, paramagnetism, ferrimagnetism, antiferromagnetism
Ferromagentism
Exchange energy
Applications of hard and soft magnets
N. Newman, [email protected], Lecture 1, page 3
A Brief Introduction to Solid-State Physics
The physical, electronic and chemical properties of solids are very diverse. Metals make up a class of non-transparent material which are thermally and
electrically conductive. In contrast, insulators are typically transparent in the visible spectral range and non-conducting. Semiconductors make up the third class
of material which have properties that vary widely. In this course, we will investigate the physical models which have been developed to understand the
properties of solids, with a strong emphasis on semiconductors.
The physical models which are currently used to predict the properties of solids use a number of approximations. Because the magnitude of the
resulting errors can not be precisely determined, the validity of these theories can be only tested by comparison of their predictions to experiment. In this
course, we will study both the physical models and the approximations. An emphasis will be placed on assessing the accuracy of each model’s predictions
when compared to experimental measurements.
Comparison to predictions of the gas phase: the accuracy of the predictions for solids is poor compared to those for the gas phase. To illustrate this
point, we examine a few theoretical predictions of gas properties.
Atomic hydrogen generated in a H2 plasma has an absorption and emission spectrum which is described by the simple Bohr model. The accuracy of
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the energy level predictions agrees with experiment within one part in 10 . Improvements using quantum electrodynamics including relativistic effects, fine
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structure and higher order corrections give predictions accurate to 1 part in 10 , the magnitude of experimental error [Experiments in Modern Physics, Chapt. 2,
A. C. Melissinos, 1966]. The properties of atoms and molecules in low pressure plasma (i.e. ,~1 mT) do not require corrections due to the presence of other
nearby gas molecules. In contrast, the energy differences between states of crystalline semiconductors can not be predicted to better than 20%. This level of
accuracy has only recently been achieved by Louie and co-workers [1993].
In general, gasses are found to accurately follow ideal gas lows under almost all experimental conditions. Deviations at very high as pressure are
found because of the finite volume of atoms and molecules and the attraction/repulsion between the atoms and molecules. For example, the Joule-Thomson
effect refers to a phenomena which occurs when a high pressure gas is injected through a small orifice into a chamber at low pressure. An idea gas is not
expected to change its temperature. However, the effects of the mutual attraction decreases when going from high pressure into the low pressure region and
energy is absorbed, resulting in cooling of the gas [Statistical Physics, Landau and Lifshitz, p. 56, 229, 1980].
The many electron problem for solids: In liquids, amorphous solids and crystalline solids, the toms are typically separated by only ~2 – 3 Angstroms (i.e., 2-3 x
-10
10 m). For this reason, the interaction between electrons from nearby atoms lays a dominant role in determining many of the material properties. This is in
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-3 1/3
contrast to the ~34 Angstrom [=(22.4x10 cm / 6.02x10 cm ) ] separation between atoms in atmospheric gas at room temperature.
Crystalline solids can be readily modeled because inside the ordered structure every atom has identical surroundings. Each atom can be considered a
“building-block”. When a building-block is modeled along with its interaction with the neighboring cells, all of the solid’s properties can be determined. For most
solids, first-principle predictions of “ground-state” properties including the density of the solid, the binding energy of the solid and the bulk modulus are
extremely accurate (<2%). These calculations use the “one-electronic approximation”, an assumption that each electron is only influenced by an “average”
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interaction with other electrons. This approximation is required to simplify the complexity of a calculation involving the interaction of ~10 /cm valence
electrons. Properties which involve excited states (e.g. bandgap and optical absorption) can not yet be calculated with similar accuracy (often with errors>
10%). Many other electronic properties including those related to superconductivity need an improved theory which takes into account interactions between
electrons and the lattice. In addition, the understanding of crystalline solids with strong electron-electron interactions (e.g. nickel oxide, ferromagnets and the
recently-discovered high temperature ceramic superconductors) are still largely not understood.
In liquids and amorphous solids, the atoms are not located periodically, making calculations difficulty. For this reason, the fundamental understanding
of crystalline solids is significantly more advanced than in liquids and amorphous materials due to their relative simplicity.
N. Newman, [email protected], Lecture 1, page 4
Questions on material’s properties
1. Why do metals reflect light and have that common grayish color,
except Au, Cu and TiN? What’s up with Au, Cu and TiN?
2. Why is snow white, yet ice is clear? How do they make white
paint?
3. Since the dielectric constant of solids is greater than 1, can
electrons in a solid go faster than the speed of light?
4. The effective mass of an electron in GaAs is 0.068 mo. Does the
electron really weigh less than mo (9.1x10-31 g)? Can you explain
why?
5. What is black gold?
6. How come solids that are electrically conductive are also
thermally conductive and optically opaque?
7. What gives precious gems like amethyst, rose quartz, citrine,
sapphire, blue diamonds and yellow diamonds their color?
Contd…
N. Newman, [email protected], Lecture 1, page 5
8. Can you make
and transparent?
a
material
that
is
both
conductive
9. Why does diamond have a high melt temperature (>3000 C), yet
NaCl doesn’t (~1000 C), despite the fact that they have similar
heats of formation?
10. How thin do you need to make Au, Al, Si or GaAs so that they
transmit a significant amount of light?
11. Why is beer yellow, yet its foam is white? Why is beer yellow?
12. What limits the storage density of CDs and DVDs?
13. Does light absorption arise from an electron’s wave-like
properties? Can a free electron in empty space absorb light?
14. How high a velocity can a conduction electron in a solid be
accelerated to? Does this result from the e-’s wave-like property?
N. Newman, [email protected], Lecture 1, page 6
Semiconductor properties relevant to solid-state devices
Bandgap
Dielectric constant
Carrier mobility, saturation velocity, lifetime and diffusion length
Carrier effective mass and density of states for VBM and CBM
Carrier effective mass and density of states for nearby band extrema
Band offsets
Energy level of native defects and impurity defects
Radiative/non-radiative lifetimes
Energy level of surface and interface states
Electron affinity, Work functions
Bandgap narrowing by high defect densities
Lattice constant, thermal expansion coefficients
Optical and acoustic phonon energies, Debye temperature
Questions:
1. What is the basic physical mechanism which determines each property.
2. Which of the properties be accurately predicted from theory?
3. If accurate predictions are not possible, can trends be understood?
N. Newman, [email protected], Lecture 1, page 7
Hall Coefficients Of Selected Elements In Moderate To High Fields
The application of an electric field combined with a perpendicular magnetic
field is used to determine the charge of current carriers in a solid.
The Hall coefficient, RH= -1/nec, is 1 for electrons.
Metal
Valence
-1/RHnec
Li
1
0.8
Na
1
1.2
K
1
1.1
Rb
1
1.0
Cs
1
0.9
Cu
1
1.5
Ag
1
1.3
Au
1
1.5
Be
2
-0.2
Mg
2
-0.4
In
3
-0.3
Al
3
-0.3
The alkali metals obey the Drude model for electrons in solids reasonably well, the
noble metals (Cu, Ag, Au) less well, and the remaining entries are inconsistent with
predictions.
From: Solid State Physics, Ashcroft and Mermin, p.15.
N. Newman, [email protected], Lecture 1, page 8
N. Newman, [email protected], Lecture 1, page 9
Photon
Energy
10
Neutron
Energy
0.0001
Electron
Energy
1
100
1K
10K
0.001
0.01
0.1
1
10
100
1K
10K
Energy (in eV)
Wavelength verses particle energy for photons,neutrons and electrons
N. Newman, [email protected], Lecture 1, page 10
100K
The problem of quantum theory centers on the fact that the particle picture and the
wave picture are merely two different aspects of one and the same physical reality.
Werner Heisenberg, The Physical Principles of the Quantum Theory, The University of
Chicago Press, p. 177, 1930.
Anyone who is not shocked by quantum theory has not understood it.
Niels Bohr, quoted by N. C. Panda in Maya in Physics, Motilal Bonarsidass Publishers,
Delhi, p. 73, 1991.
I think that I can safely say that nobody understands quantum mechanics.
Richard Feynman, The Character of Physical Laws, British Broadcasting Corporation,
London, p. 129, 1965.
If this [quantum theory] is correct, it signifies the end of physics as a science.
Albert Einstein, quoted by L. I. Ponomarev in The Quantum Dice, IOP publishing, Bristol, p.
80, 1993.
Exact science of the last thirty years derives its special significance from the fact
that its different branches, i.e., Astronomy, Physics and Chemistry have been
followed back to their common root-atomic physics.
Werner Heisenberg, Philosophical Problems of Nuclear Science, Faber and Faber
Publishers, London, p. 27, 1952.
N. Newman, [email protected], Lecture 1, page 11
Wave-like
Light:
Matter:
Particle-like
l = n/c
E= h n
2
2
E = mv /2= p /2m
l = h/p
Time-independent Schrodinger equation:
2
2
2
-ħ /2m d y/dx + V y = Ey
Time-dependent Schrodinger equation:
2
2
2
-ħ /2m d y/dx + V y = iħ dy/dt
i[kx-wt]
y=e
with v = w/k
k = 2p/l
N. Newman, [email protected], Lecture 1, page 12
Interaction between
electrons (w/ wave properties) and ions
Case 1:
k = 0; l = 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
ions
Case 2:
k = p/a; l = 8a
+
+
ions
Case 3:
k = p/2a; l = 4a
+
+
+
ions
N. Newman, [email protected], Lecture 1, page 13
+
+
+
N. Newman, [email protected], Lecture 1, page 14
Properties of Free Electrons
How do we use wave-like properties to describe
an electron?
p= mv = k
Kinetic energy
E = 1/2 mv2
= p2/2m
= 2k2/2m
Y = e i(kx-wt)
l = 2p/k
p = -i d/dx
p Y = -i d /dx Y
p Y = kY
p = k
= mv
0
1s
2s
3s
Momentum
p= mv= k
v = dE /d (k) = Dk/m=p/m
N. Newman, [email protected], Lecture 1, page 15
Can a free electron absorb light and still conserve both
energy and momentum?
J
Kinetic energy
E= 1/2 mv2
= p2/2m
= 2k2/2m
0
dE/dk = 2k/m = v
dE2/dk2 =  2/m
7.2 x 109m-1
k
velocity
v =  k/m
5.4 x 106m/s
l = 2p/k
0
N. Newman, [email protected], Lecture 1, page 16
k
7.2 109m-1
In an electric field (F= -q E = ma)?
How much current do free electrons carry
in an electric field (F= -qE = ma)?
J = qnv= -q2nEt/m
k
Kinetic energy
30ps
E = 1/2 mv2
= p2/2m
= 2k2/2m
1.3 *10-17 J
5.4 *106 m/s
3.6 *106 m/s
20ps
1.8 *106 m/s
F = dp/dt
D p = FDt
D k(t) = -eEDt
(electric field)
10ps
V=1.8x106 m/s
0
Momentum
p= mv= k
7.2 x 109 m-1
E = 1V/1m = 106 V/m
V = qEt/m = ((1.6 x 10-19e)(106 V/m) * t /(9.1.10-31 kg)
= 1.8 x 1017 m/s2 * t
p = mv = 1.6 x 10 -13 kg* m/s * t
N. Newman, [email protected], Lecture 1, page 17
Now, we will try to understand how to model a localized
entity, such as an electron that have both wave-like and
particle-like properties.
First, lets take a wave with a wavelength, l , of p/2
x(m)
N. Newman, [email protected], Lecture 1, page 18
Now,if we add three waves with similar wavelength,
y(x) := (sin(1.9*x) + sin(2*x) +sin(2.1*x))/3
Dk = 0.2 m-1
Now,if we add five waves with similar wavelength,
y(x) := (sin(1.9*x) + sin(1.95*x) + sin(2*x) + sin(2.05*x) + sin(2.1*x))/5
Dk = 0.2 m-1
N. Newman, [email protected], Lecture 1, page 19
Now,if we add 10 waves with similar wavelength,
number := 10
number
y(x) :=( 
L=0
Sin[ (1.9 + 0.2L / number) * x ] ) / (number+1)
Dk = 0.2 m-1
Now,if we add 20 waves with similar wavelength,
number := 20
number
y(x) :=(  Sin[
L=0
(1.9 + 0.2L / number) * x ] ) / (number+1)
Dk = 0.2 m-1
N. Newman, [email protected], Lecture 1, page 20
Now lets look at the range in k
y(x) :=(sin(1.95*x) + sin(2*x) +sin(2.05*x))/3
Dk = 0.10 m-1
Here y(x) := (sin(1.9*x) + sin(2*x) +sin(2.1*x))/3
Dk = 0.20 m-1
N. Newman, [email protected], Lecture 1, page 21
Here y(x) := (sin(1.8*x) + sin(2*x) +sin(2.2*x))/3
Dk = 0.30 m-1
Dk
*
Dx
N. Newman, [email protected], Lecture 1, page 22
1
N. Newman, [email protected], Lecture 1, page 23
N. Newman, [email protected], Lecture 1, page 24
eV
0.1
X
0
- 0.1
a/
2
a
3/
2a
f(x) = -0.1(2c/a + 2/p S (-1)n /n * sin(npc/(a/2))*cos(npx/(a/2))eV
n=1
N. Newman, [email protected], Lecture 1, page 25

Step # 1
The free electron  vs.  parabola in one
dimension

0

Step # 2
-p
- p/a
p/a
0
2p/a
Here (k) = (k +K) where K is 2p/a

Step # 3
3p/a
2p

When lelectron = l lattice
lelectron = 2p/k and
u2
k = p/a
Periodicity of Lattice = 2a
u1
- p /a
0
p/a

N. Newman, [email protected], Lecture 1, page 26
Step # 4

-
3p/
2a
-
2p/
a
-
p/
a
0
N. Newman, [email protected], Lecture 1, page 27
p/
a
2p/
a
3p/

2a

Step # 5
- p/a
0
p/

a
All the information for this figure is from -p/a to p/a
N. Newman, [email protected], Lecture 1, page 28
For k far away from p/a; there is a near-zero net average interaction.
Case 1:
k = 0; l=
ions
Case 2:
k = p/a; l = 8a
ions
Case 3:
k = p/2a; l = 4a
ions
N. Newman, [email protected], Lecture 1, page 29
N. Newman, [email protected], Lecture 1, page 30
N. Newman, [email protected], Lecture 1, page 31
When lelectron = 2a and k = p/a
Bonding:
|y|2 maximum between repulsive ion cores
|y|2 minimum at repulsive ion cores
ions in core electrons
lelectron
a
l electron = 2a
When lelectron = 2p/k = 2a; i.e. k =p/a
Anti-Bonding:
|y|2 minimum between repulsive ion cores
|y|2 maximum at repulsive ion cores
ions in core electrons
a
l electron = 2a
N. Newman, [email protected], Lecture 1, page 32
lelectron
N. Newman, [email protected], Lecture 1, page 33
Bonding:
y = u1y1 +u2 y2
y1
v2
y2
E = < y|H| y>
v2
< y| y>
E=
u1u1H11 + u1u2H12+u2u1H21+u2u2H22
u1u1 < y1 | y1 > + u1u2 < y1 | y2 > + u2u1 < y2 | y1 >+u2u2 <
y2 | y2 >
Find u1 and u2 that minimize the energy
dE
du1
dE
du2
=0
=> H11u1 + H12u2 = E u1
=0
=> H21u1 + H22u2 = E u2
H11 H12
u1
H21 H22
u2
E = (H11 + H22)/2 +
=E
u1
u2
(H11 - H22) 2/ 4 + H21* H12
If H21, H12 are small compared to H11-H22,
then E = H11 +(( H21. H12)/(H11 - H22)) and U2 = (H21 /(H11 - H22) )* u1
N. Newman, [email protected], Lecture 1, page 34
<
y1 |H | y1 > = H11
<
y2 |H | y2> = H22
< y1| y1 > = < y2 | y2 > = 1
< y1 | y2 >= < y2 | y1 > = 0
N. Newman, [email protected], Lecture 1, page 35
N. Newman, [email protected], Lecture 1, page 36
Movies of energy levels of
Bravis lattice will be supplied
by the instructor
• Energy
levels of Bravis Lattice
• Extension
of the energy levels of
Bravis Lattice
N. Newman, [email protected], Lecture 1, page 37
N. Newman, [email protected], Lecture 1, page 38