Friday, March 17, 2023

What Is K Space In Solid State Physics

A Lattice With Two Types Of Atoms

Introduction to k-Space

Now the solutions are:

Connected with each value of \ are two values of \, as can be seen in the graph. The upper line describes the optical branch, the lower line the acoustical branch. In the optical branch, both types of ions oscillate with opposite phases, in the acoustical branch they oscillate with the same phase. This results in a much larger induced dipole moment for optical oscillations, and also a stronger emission and absorption of radiation. Furthermore each branch has 3 polarization directions, one longitudinal and two transverse.

Plane Waves And Reciprocal Space

The complex exponentials such as those that are used for a Fourier series are often called plane waves. A plane wave has the form,

Here $\vec = k_x\,\hat+k_y\,\hat+k_z\,\hat$ is a vector called a wave vector. Consider the value of the plane wave at a particular point $\vec$. In general, the value at this point is some complex number. Now move away from that point in a direction perpendicular to $\vec$ to $\vec+\vec_$. Every point on the plane that passes through $\vec$ and is perpendicular to $\vec$ has the same value, $e^\cdot\left} = e^\cdot\vec}$.

If you move in a direction parallel to $\vec$, the function oscillates with a wavelength $\lambda = 2\pi/|\vec|$. The real parts of some plane waves are plotted below. You just have to imagine the imaginary parts.

 $\vec=6\,\hat$ $\vec=6\,\hat$ $\vec=4\,\hat+4\,\hat$ $\vec=3\,\hat-5\,\hat$

Q.1. Is there any difference between solid-state physics and condensed matter physics?

Answer Yes there is a difference between them. Like, condensed matter physics focus on the commonality of scientific problems faced by physicists who work on solids, liquids, plasmas, and other complex matter, while solid-state physics is often in association with restricted industrial applications of metals and semiconductors.

Q.2. What is a unit cell in solid-state physics?

Answer A unit cell is the most basic and repeating structure of any solid. It is the least volume consuming structure of a solid. When the unit cell repeats itself, it forms a network, called a lattice.

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How Many Electrons Are In The $k$

Let’s assume we have a metal cube with edge length $L$ and in its volume $L^3$ are $N$ conduction electrons. The components of the wave vector are quantized. i.e. $k_x=m\cdot$ where $m=0,\pm 1,\pm 2,\dots$.

Question: How many electrons can be in a volume element of the $k$-space $\triangle k_x \cdot\triangle k_y\cdot \triangle k_z=^3$?

If the electrons are free, the single particle excitation spectrum is the well-known:$$E_ = \frac$$where $\vec$ is referred to the unique value of the wave-vector that exists in a unit cell.By the Pauli exclusion principle, you know that you can have just one electron in a given quantum state, that, in this case, is labelled by the wave vector and the spin of the electron:$$|\vec,S_ \rangle$$where $S_$ is the projection of the spin along the $z$ direction.This means that you can have two electrons in a single unit cell labeled with a wave-vector $\vec$, beacuase of the spin degeneracy that in this case is:$$= 2$$where $S$ is the modulus of the spin that for an electron is $S=\frac$. Let’s summarize: for a given wave vector $\vec$, and so, for a fixed cell of the Fourier space, you can have two electrons with the same energy , described by the quantum states:$$|\vec,S_ \rangle = |\vec,\frac \rangle$$$$|\vec,S_ \rangle = |\vec,-\frac \rangle$$This means that the answer to your question is: you can have at maximum 2 electrons per unit-cell and so you can have: two, one, or zero electrons.

Types Of Condensed Matter Cohesion

• How orbitals fill in elements, resulting in the shape of the periodic table
• Ionic bonding makes salts
• Charge is transferred between different species, and then ions electrostatically attract.
• Ionization potential and Electron affinity
• Long range interaction requires consideration of more than nearest neighbors to get accurate binding energy
• Non-directional interaction leads to moderately high densities, but not too high because can’t have triangular coordination
• Typical binding energies 2.55eV/ion high melting temps
• Covalent bonding involves bonds, as in a molecule
• Existence of bonds is a purely quantum mechanical phenomenon. Having multiple atom nuclei allows electrons the flexibility to find lower-energy wave functions
• Bonds are directional, often leading to low densities/packing fractions
• Typical energies 35eV/ion high melting temps , although many chemically react first
• Metallic bonding occurs when some electrons cease to be localized in atoms or bonds
• Non-directional interaction leads to high densities
• Typical binding energies 13eV/ion moderately high melting temp
• Hydrogen bonding: a sort of a permanent dipole-permanent dipole attraction
• Only hydrogen creates dipoles this strong, when covalently bound to other atoms, due to having only a single electron
• Typical binding energies 0.1eV/Hydrogen
• van der Waals force binding makes molecular crystals
• fluctuating dipole-dipole attraction varies as 1/r6 often modeled with Lennard-Jones 6-12 potential
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Significance Of Band Gap In K

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ZapperZ said:There is something weird about that diagram. is an energy transition, but the lateral transition is a MOMENTUM transition, not energy. So the units/dimensions are all wrong.

NaOH said:Let the direct band gap be denoted as g. Let’s say the temperature is such that it is possible for energies up to . But since )< , can I say that I will only observe appreciable current when I apply a electric field in the direction of +kr?

Why This Is The Density Of Points In $k$

I’m reading a solid state physics book and there’s something which is confusing me, related to the free electron gas.

After solving Schrodinger’s equation with $V = 0$ and with periodic boundary conditions, one finds out that the allowed values of the components of $\mathbf$ are:

$$k_x = \dfrac, \quad k_y=\dfrac, \quad k_z = \dfrac.$$

In the book I’m reading the author says that it follows from this that: there is one allowed wavevector – that is, one distinct triplet of quantum numbers $k_x,k_y,k_z$ – for the volume element $^3$ of $\mathbf$ space.

After that he says that this implies that in the sphere of radius $k_F$ the total number of states is

$$2 \dfrac=\dfrack_F^3 = N,$$

where the factor $2$ comes from spin.

Now, why is that the case? Why it follows from the possible values of $k_x,k_y,k_z$ that density of points in $k$-space? I really can’t understand this properly.

Consider $k_x$, $k_y$, and $k_z$ defining the three orthogonal axes of a three dimensional space. This is what he calls $\boldsymbol$ space.

The allowed values of $\boldsymbol$, that is $k_i = 2 n_i \pi / L$, are represented by evenly spaced points in this $\boldsymbol$ space. Each allowed point is separated from its closest neighbours by a distance $2\pi/L$ along each axis. To see this, just compute the separation between points with consecutive integers $n_i$ along each axis:

$$\frac – \frac = \frac$$

Therefore there is a $\boldsymbol$ space volume of $^3$ for each allowed point.

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What Is Solid State Physics

One definition of Solid-state physics is, its the study of the physical properties of solids in terms of basic physical laws. Like the study of the electrical, dielectric, magnetic, elastic, and thermal property of solids. Solid-state physics is more like chemistry than another branch of physics because it focuses on common properties of huge classes of materials. The study of Solid state physics focusses on how physics properties are in link to the electronic structure. Weve retained the term Solid State Physics, but Albeit condensed matter physics is more commonly used.

Condensed Matter Physics includes liquids and non-crystalline solids like glass, which we shall not discuss intimately. During the late thirties and forties, modern Solid State Physics came in light. It had its most extensive expansion with the formation of the transistor, integrated circuits, and microelectronics.

Presentation On Theme: Fourier Transform From R To K: A = A E I K R D 3 R Inverse Ft From K To R: A = 3 A E +i K R D 3 K X

Solid State Physics in a Nutshell: Week 2.1 Lattice and Basis

1 Fourier transform from r to k: A = A e i k r d 3 r Inverse FT from k to r: A = 3 A e +i k r d 3 k X-rays scatter off the charge density , neutrons scatter off the spin density. Coherent scattering creates the Fourier transform of from real to reciprocal space: A = e i r d 3 r A = |A| e i I = |A| 2 = k k 0 = scattering vector = G hkl for periodic structures General theory of diffraction measured intensity I Real space Reciprocal space k0k0 k

2 Structure determination by diffraction The diffraction pattern is determined by three considerations: 1)The Bragg condition determines the position of the diffraction spots in k-space. It originates from the whole crystal lattice. 2)The structure factor describes the intensity modulation of the diffraction spots by the atoms inside the unit cell. 3)The atomic scattering factor describes the diffraction at an individual atom. This is in tune with the mantra that large objects in real space correspond to small objects in k-space. 1) represents the largest possible object in real space , and it becomes the smallest possible object in k-space . 2) represents a medium-sized object , and 3) the smallest object . They affect increasingly larger portions of k-space.

6 Laue diffraction pattern Laue diffraction pattern of NaCl taken with neutrons. See a projection of k-space.

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Semiconductor Physics: Density Of States

To calculate various optical properties such as the rate of absorption oremission and how electrons and holes distribute themselves within a solid, weneed to know the number of available states per unit volume per unit energy.We first calculate the available states in k-space and then use theenergy-momentum relation in parabolic bands to give the density of states interms of energy.

By considering the electrons in a solid as a free electron gas, that is, theelectrons are free to wander around the crystal without being influenced by thepotential of the atomic nuclei, we can obtain a relationship for the number ofavailable states in a solid. A free electron has a velocity v and a momentump=mv. Its energy consists entirely of kinetic energy therefore,

Prince Louis De-Broglie, hypothesised that if waves could exhibit particle-likeproperties, then might particles also exhibit wave-like properties? This ideais expressed as particle-wave duality and allows us to give the electron a wavenumberk.

In this way, the electron can be represented by a vector in velocity, momentumor k-space. If we choose to represent the electron state as a vector, it pointsin a direction given by the components magnitude of the basis vectors ink-space. It should be apparent that vectors of the same magnitude have the sameenergy forming spherical shells. This can be understood better, if we considerthe equation for the energy of the electron in terms ofk.

where nx, ny, nzare integers.

Points Of Symmetry In $k$

Can you relate a point in the reciprocal space with a vector in real space?

How do I find the family of planes that represent a point of symmetry in the Brillouin zone?

For example, germanium has its characteristic gap at the center of the first Brillouin zone, that is, at the gamma point. Which will be the family of planes that will present this characteristic gap?

The gamma point represents waves with $k=0$, infinite wavelength. In the tight-binding approximation, this means a constant value of the phase factor for the atomic orbitals.

Germanium is not an easy example. It has an indirect band gap. But many salts have direct band gaps at the gamma point, for example MgO. In a tight-binding picture, at $\Gamma$ the phase factors are the same throughout the crystal. The lower band corresponds to electron wave functions with mostly oxygen $2p$ character, the upper band to wave functions with mainly magnesium $3sp$ character.

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Vibrational Modes Of Solids

Equations and derivations:

• Force on an atom in a 1D chain of atoms & springs, and equations of motion from Newton’s law
• Wave ansatz, exponential form
• Dispersion relation for 1D chain of atoms

Question types:

• Finding dispersion relations from equations of motion
• More finding densities of modes

Concepts:

• Born-Oppenheimer approximation: because electrons are lighter and faster, they rapidly equilibrate to whatever motion the ions make. Therefore, we can think of the electrons as providing an effective force between the ions . We do not have to consider the electrons as separate things they just move with the ions.
• 1D chain of identical ions, assuming nearest neighbor forces
• As long as ion displacements are not large, the forces between them will be approximately quadratic in position: Hookes Law! based on Taylor expansion of the force
• Plane waves satisfy these equations of motion as long as the appropriate dispersion relation is obeyed
• Due to aliasing, only k‘s in the 1st Brillouin zone are physically useful. By counting degrees of freedom, we know that these plane wave solutions are a complete set.
• A Lattice With One Type Of Atoms

In this model for crystal vibrations only nearest-neighbour interactions are taken into account. The force on atom \ with mass \ can then be written as:

Assuming that all solutions have the same time-dependence \\) this results in:

Further it is postulated that: \\exp\).

This gives: \\). Substituting the later two equations in the first results in a system of linear equations, which only has a solution if their determinant is 0. This gives:

Only vibrations with a wavelength within the first Brillouin zone have a physical significance. This requires that \.

The group velocity of these vibrations is given by:

and is 0 on the edge of a Brillouin Zone. There, there is a standing wave.

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Crystal Structure And Properties

Many properties of materials are affected by their crystal structure. This structure can be investigated using a range of crystallographic techniques, including X-ray crystallography, neutron diffraction and electron diffraction.

The sizes of the individual crystals in a crystalline solid material vary depending on the material involved and the conditions when it was formed. Most crystalline materials encountered in everyday life are polycrystalline, with the individual crystals being microscopic in scale, but macroscopic single crystals can be produced either naturally or artificially.

Real crystals feature defects or irregularities in the ideal arrangements, and it is these defects that critically determine many of the electrical and mechanical properties of real materials.