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Electrical, optical and magnetic properties of materials 7/13/2010 Hand-out #1

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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.

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.

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

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 the energy level predictions agrees with experiment within one part in 105. Improvements using quantum electrodynamics including relativistic effects, fine structure and higher order corrections give predictions accurate to 1 part in 108, 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 10-10m). For this reason, the interaction between electrons from nearby atoms lays a dominant role in determining many of the material properties. This is in contrast to the ~34 Angstrom [=(22.4x103 cm3 / 6.02x1023 cm-3)1/3] 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” interaction with other electrons. This approximation is required to simplify the complexity of a calculation involving the interaction of ~1023/cm3 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.

Questions on material’s properties

- Why do metals reflect light and have that common grayish color, except Au, Cu and TiN? What’s up with Au, Cu and TiN?
- Why is snow white, yet ice is clear? How do they make white paint?
- Since the dielectric constant of solids is greater than 1, can electrons in a solid go faster than the speed of light?
- 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?
- What is black gold?
- How come solids that are electrically conductive are also thermally conductive and optically opaque?
- What gives precious gems like amethyst, rose quartz, citrine, sapphire, blue diamonds and yellow diamonds their color?
- Contd…

Can you make a material that is both conductive and transparent?

- 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?
- How thin do you need to make Au, Al, Si or GaAs so that they transmit a significant amount of light?
- Why is beer yellow, yet its foam is white? Why is beer yellow?
- What limits the storage density of CDs and DVDs?
- Does light absorption arise from an electron’s wave-like properties? Can a free electron in empty space absorb light?
- How high a velocity can a conduction electron in a solid be accelerated to? Does this result from the e-’s wave-like property?

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?

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.

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.

Photon 10 100 1K 10K 100K

Energy

Neutron 0.0001 0.001 0.01 0.1 1

Energy

Electron 1 10 100 1K 10K

Energy

Energy (in eV)

Wavelength verses particle energy for photons,neutrons and electrons

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.

Light:l = n/c E= hn

Matter:l = h/p E = mv2/2= p2/2m

Time-independent Schrodinger equation:

-ħ2/2m d2y/dx2 + Vy = Ey

Time-dependent Schrodinger equation:

-ħ2/2m d2y/dx2 + Vy = iħ dy/dt

y = ei[kx-wt]

with v = w/k

k = 2p/l

electrons (w/ wave properties) and ions

Case 1:

k = 0; l =

ions

ions

ions

Case 2:

k = p/a; l = 8a

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

Case 3:

k = p/2a; l= 4a

+

+

+

+

+

+

+

+

+

How do we use wave-like properties to describe an electron?

p= mv = k

Kinetic energy

E = 1/2mv2

= p2/2m

= 2k2/2m

Y = e i(kx-wt)

l = 2p/k

0 1s 2s 3s

- p = -id/dx
- p Y = -id /dx Y
- p Y = kY
- p = k

= mv

Momentum

p= mv= k

v = dE /d (k) = Dk/m=p/m

Can a free electron absorb light and still conserve both energy and momentum?

J

Kinetic energy

E= 1/2mv2

= p2/2m

= 2k2/2m

0 k

7.2 x 109m-1

dE/dk = 2k/m = v

velocity

5.4 x 106m/s

dE2/dk2 = 2/m

v = k/m

l = 2p/k

k

0

7.2 109m-1

In an electric field (F= -q E = ma)?

Kinetic energy

E = 1/2mv2

= p2/2m

= 2k2/2m

How much current do free electrons carry

in an electric field (F= -qE = ma)?

J = qnv= -q2nEt/m

k

30ps

1.3 *10-17 J

5.4 *106 m/s

3.6 *106 m/s

20ps

1.8 *106 m/s

F = dp/dt

10ps

10ps

V=1.8x106 m/s

0

D p = FDt

D k(t) = -eEDt

(electric field)

Momentum

p= mv= k

7.2 x 109 m-1

- E = 1V/1m = 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

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

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

Now,if we add 10 waves with similar wavelength,

number := 10

number

y(x) :=( Sin[ (1.9 + 0.2L / number) * x ] ) / (number+1)

L=0

Dk = 0.2 m-1

Now,if we add 20 waves with similar wavelength,

number := 20

number

y(x) :=( Sin[ (1.9 + 0.2L / number) * x ] ) / (number+1)

L=0

Dk = 0.2 m-1

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

The free electron vs. parabola in one dimension

Step # 2

0

-p

p/a

3p/a

- p/a

2p

2p/a

0

Here (k) = (k +K) where K is 2p/a

Step # 3

When electron = lattice

electron = 2p/k and

k = p/a

Periodicity of Lattice = 2a

u2

u1

- p /a

p/a

0

For k far away from p/a; there is a near-zero net average interaction.

Case 1:

k = 0; l=

Case 2:

k = p/a; l = 8a

ions

ions

ions

Case 3:

k = p/2a; l= 4a

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

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

lelectron = 2a

a

lelectron

lelectron = 2a

y2

v2

v2

Bonding:

y = u1y1 +u2y2

=<y|H| y>

<y| y>

= u1u1H11 +u1u2H12+u2u1H21+u2u2H22

u1u1 <y1 | y1 > + u1u2 <y1 | y2 >+u2u1 <y2 | y1 >+u2u2 <y2 | y2 >

Find u1 and u2 that minimize the energy

d => H11u1 +H12u2 = E u1

du1

d => H21u1 +H22u2 = E u2

du2

H11 H12 u1 =u1

H21 H22 u2 u2

E = (H11 +H22)/2 + (H11 - H22)2/ 4 + H21* H12

If H21, H12 are small compared to H11-H22,

then = H11 +((H21. H12)/(H11 - H22)) and U2 = (H21 /(H11 - H22) )* u1

= 0

= 0

< y1|H |y1 > = H11

< y2|H |y2> = H22

< y1| y1 > = < y2 | y2 > = 1

< y1 | y2 >= < y2 | y1 > = 0

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

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