Solid State Physics Bands & Bonds. PROBABILITY DENSITY The probability density P(x,t) is information...

Solid State PhysicsSolid State Physics

Bands & BondsBands & Bonds

PROBABILITY PROBABILITY

DENSITYDENSITY

The probability density P(x,t) is information that tells us something about the likelihood of obtaining one of a variety of possible results in a position measurement.

As particles move through space, their wavefunction evolves with time and the probability density changes (along with the position information it contains).

The quantum world is governed by statistics.

Postulates ofPostulates ofQuantum MechanicsQuantum Mechanics1. Every physically-realizable state of a system is

described by a state function that contains all accessible physical information about the system in that state.

2. If a system is in a quantum state , then is the probability that in a position measured at time t the particle will be detected in an infinitesimal volume dv.

3. In quantum mechanics, every observable is expressed by an operator that is used to obtain physical information about the observable from state functions. For an observable from state functions.

Postulates ofPostulates ofQuantum MechanicsQuantum Mechanics

4. The time development of the state functions of an isolated quantum system is governed by the time- dependent Schrodinger equation (TDSE).

Classical versus QuantumClassical versus Quantum

The TDSE is the “equation of motion” in quantum mechanics just as Newton’s 2nd is the equation of motion in classical mechanics.

Classical versus QuantumClassical versus Quantum

In classical physics a macroscopic particle moves through 2D space along a trajectory.

In quantum physics a microscopic particle moves through 1D space and is represented by a complex state function, the real part of which is shown.

TranslationTranslation

Under translation, a macroscopic free particle must not alter its probability density, even though the state function will be altered.

Wave Packets

To construct a wave packet with a single region of enahnced amplitude, we must superpose an infinite number of plane wave functions with infinitesimally differing wave numbers.

( )1( , ) ( )

2i kx tx t A k e dk

Note: although this is developed for the free particle, it is applicable to many one-dimensional quantum systems.

1( ,0) ( )

2ikxx A k e dk

The amplitude function A(k) is the Fourier transform of the state function at time t = 0.

Time-independent Time-independent SchrödingerSchrödingerIn solid state physics, we’ll use the time-independent equation because in the study of most material properties solids are modeled assuming stationary nuclei at their equilibrium positions in the solid and other properties are statistically uniform.

2

2

( )( ) ( ) ( )

2

d xV x x E x

m dx

Particle in a BoxParticle in a Box

L0

EnergiesEnergies

Why do atoms bond to each other?Why do atoms bond to each other?

When they are bound together, the energy When they are bound together, the energy of the system is lower than when the of the system is lower than when the atoms are isolated.atoms are isolated.

In fact, most electrons occupy energy In fact, most electrons occupy energy states that are lower than the ground state states that are lower than the ground state of an isolated atom.of an isolated atom.

Consider the available energies for electrons in the materials.

As two atoms are brought close together, electrons

must occupy different energies due to Pauli Exclusion principle.

Instead of having discrete energies as in the case of free atoms,

the available energy states form bands.

Conductors, Insulators, and Semiconductors

Wave Equation for an Electron in a Periodic Potential

We’ve been considering electrons as moving freely in a box potential. In reality, the periodicity of the crystal leads to a periodic potential.

Bonding MechanismsBonding Mechanisms

CovalentCovalent Electron concentration is greater in the neighborhood Electron concentration is greater in the neighborhood

along the line “joining” two atoms together.along the line “joining” two atoms together. MetallicMetallic

Electron concentration is uniform throughout the Electron concentration is uniform throughout the spaces in between atoms.spaces in between atoms.

IonicIonic An electron can be transferred from the neighborhood An electron can be transferred from the neighborhood

surrounding one atom to that of another and the surrounding one atom to that of another and the atoms can then be viewed as bonded through atoms can then be viewed as bonded through Coulomb interaction.Coulomb interaction.

Can be qualitatively distinguished on the basis of electron distribution.

Quantum Mechanical ViewQuantum Mechanical View

2

2

2

2

2

22

2

),r(),r(),r(),r( where

),r(),r()r(),r(

2

z

t

y

t

x

tt

t

titUt

m

Calculation of the total energy of a solid begins with Calculation of the total energy of a solid begins with finding solutions to the Schrödinger equation for the finding solutions to the Schrödinger equation for the electron energies and wavefunction.electron energies and wavefunction.

Electron wave functions overlap – forming a new wave function


The wave function The wave function can be written in can be written in terms of its spatial terms of its spatial part and its temporal part and its temporal part.part.

The temporal part has The temporal part has the form…the form…

)()r(),r( tft

tietf )(

Quantum Mechanical ViewQuantum Mechanical View The real and imaginary parts of the wave The real and imaginary parts of the wave

function oscillate sinusoidally with time, but the function oscillate sinusoidally with time, but the probability density is independent of time.probability density is independent of time.

The angular frequency of the wave function is The angular frequency of the wave function is related to the energy of the electron.related to the energy of the electron.

22)r(),r( t

get... we

)r(),r( with so , tietE

)r()r()r()r(2

2 EUm


The wave function The wave function and the probability and the probability density depend on density depend on the potential the potential energy function for energy function for the electron.the electron.

The individual eThe individual e are replaced by a are replaced by a continuous continuous distribution.distribution.

Hartree ApproachHartree Approach

i i

ien

ZeU

Rr4)r(

nsinteractio nuclei-electron

0

2

ii

ee

n

dne

U

2

0

2

),r()r( where

rr

)r(

4)r(

nsinteractioelectron -electron

Hydrogen BondingHydrogen Bonding

HH22 molecule molecule

2

0

1 1

4(r)

eU

r r R

R

rr R

Proton AProton B

electron

Antibonding StatesAntibonding States

Not all molecular states result in a lowering of Not all molecular states result in a lowering of energy.energy.

StatesStates

When atoms are brought close together, an When atoms are brought close together, an electron state is created for each core state of electron state is created for each core state of an atom – some bonding (lower energy) and an atom – some bonding (lower energy) and some antibonding (higher energy).some antibonding (higher energy).

These states are all occupied. Good thing we These states are all occupied. Good thing we have the Pauli exclusion principle.have the Pauli exclusion principle.

Atoms are attracted by the outer electrons, Atoms are attracted by the outer electrons, occupying bonding states, and repelled mostly occupying bonding states, and repelled mostly by ion core electrons.by ion core electrons.

Covalent BondingCovalent Bonding

Let’s develop a bonding wave function through Let’s develop a bonding wave function through linear combination of localized atomic orbitals for linear combination of localized atomic orbitals for each atom each atom aa::

atom of each typefor different constant, a is

atom of each typefor different orbital, atomican is

states atomic allover runs

where

)r()r(

an

an

ann

ana

A

n

A


For two atoms:For two atoms:

Lots of solutions, but the lowest energy states Lots of solutions, but the lowest energy states have coefficients that make the overlap large.have coefficients that make the overlap large.

Eqn.r Schrodinge esatisfy th chosen to , , ,

positions atomic theare R,R

where

)Rr()Rr()r(

a

a

babnan

b

bbnbana

CCAA

CC


There are two types of covalent bonds : There are two types of covalent bonds : sigmasigma and and pipi. When the covalent bonds are linear or aligned . When the covalent bonds are linear or aligned along the plane containing the atoms, the bond is along the plane containing the atoms, the bond is known as sigma (known as sigma () bond. Sigma bonds are strong ) bond. Sigma bonds are strong and the electron sharing is maximum. Methane and the electron sharing is maximum. Methane CHCH44 is a good example for sigma bond and it has is a good example for sigma bond and it has

four of them.four of them.

Covalent BondsCovalent Bonds

When the electron orbitals overlap laterally, the bond is called the pi () bond. In pi bonds, the resulting overlap is not maximum and these bonds are relatively weak.

Covalent Bonds have well-defined directions in space. Attempting to alter those directions is resisted – making them both hard and brittle.

Ionic BondingIonic Bonding

Consider two different atoms, such as Ga and As joined by a covalent Consider two different atoms, such as Ga and As joined by a covalent bond. The bond. The functions come from different orbitals. The C constants functions come from different orbitals. The C constants would be different. The wave function still spreads between the would be different. The wave function still spreads between the atoms, but the probability is that the shared electron will be nearer to atoms, but the probability is that the shared electron will be nearer to one atom than another.one atom than another.

Atoms in this situation act like oppositely charged ions that are Atoms in this situation act like oppositely charged ions that are attracted by the Coulomb force.attracted by the Coulomb force.


1+1


Bonding may be evaluated energetically via Bonding may be evaluated energetically via calorimetric measurements.calorimetric measurements.

1.1. measure a-a bond energy in pure a materialmeasure a-a bond energy in pure a material

2.2. measure b-b bond energy in pure b materialmeasure b-b bond energy in pure b material

3.3. measure a-b material – This will be greater measure a-b material – This will be greater than the average of a-a and b-b bonds. than the average of a-a and b-b bonds. The difference is the energy of the ionic The difference is the energy of the ionic bond.bond.

van der Waals Bondingvan der Waals Bonding

An atom has an electric dipole moment An atom has an electric dipole moment when the average position of its electrons when the average position of its electrons does not coincide with that of the nucleus.does not coincide with that of the nucleus.

The dipole moment produces an electric The dipole moment produces an electric field, even if the atom is neutral.field, even if the atom is neutral.

This field induces dipole moments in other, This field induces dipole moments in other, nearby atoms.nearby atoms.

p1

p2

E

R


1130

pR̂R̂p31

4

1E

R


1130

2 pR̂R̂p31

4Ep

R

The potential energy of a dipole in an electric field is...

21

2162

02 p)R̂p(3

1

)4(E p

RU

GraphiteGraphite

1.42 Å

3.4 Å

A

B

A

Metallic BondsMetallic Bonds

The outer electrons of metallic atoms are loosely The outer electrons of metallic atoms are loosely bound. When a solid formed, the potential bound. When a solid formed, the potential energy barrier is substantially reduced.energy barrier is substantially reduced.

Metals typically found in columns IA, IIA, IB, IIB Metals typically found in columns IA, IIA, IB, IIB of the periodic table have cohesive energies of of the periodic table have cohesive energies of around 1 to 5 ev/atom (for comparison, around 1 to 5 ev/atom (for comparison, convalent bonds are in the 3 to 10 eV/atom convalent bonds are in the 3 to 10 eV/atom range).range).

Solid State Physics Bands & Bonds. PROBABILITY DENSITY The probability density P(x,t) is information...

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Transcript of Solid State Physics Bands & Bonds. PROBABILITY DENSITY The probability density P(x,t) is information...