Solid State Physics-II, Physics tutorial

Solid State Physics:

Solid state physics is a study of how atoms arrange themselves in solids and what properties such solids have. The atoms arrange in specific patterns as patterns minimize energy in the binding that is typically with more than one neighbor in solid. The ordered (periodic) arrangement is known as crystal a disordered arrangement is known as amorphous. All macroscopic properties such as electrical conductivity, color, density, elasticity and more are determined by and can be computed from microscopic structure.

Solid-state physicists do research in the variety of areas ranging from technologically significant work on semiconductors and magnetic materials to work of academic interest only, like in field of superconductivity.

Schrodinger Equation:

Framework of most solid state physics theory is Schrödinger equation of non relativistic quantum mechanics:

113_Schrödinger Equation.jpg

The most unexpected thing is great variety of qualitatively different solutions to Schrödinger's equation which can arise. In solid state physics you can compute all properties with Schrödinger equation, but equation is intractable and can be only solved with estimations. Model for solid is simplified until Schrödinger equation can be solved. Frequently this engages neglecting electron-electron interactions. The resulting wave function is considered as the product of hydrogen wave functions. Due to this it is possible to solve new equation exactly. Total wave function for electrons should obey Pauli Exclusion Principle. Sign of wave function should change when two electrons are exchanged.

Periodicity: Crystal Structures:

Most solid materials have crystalline structure which signifies spatial periodicity or translation symmetry.  All lattices can be attained by repetition of building block called basis. We assume that there are 3 non-coplanar vectors a1, a2, and a3 which leave all properties of crystal unchanged after shift as a whole by any of the vectors.  As a result, any lattice point R' could be attained from another point R as

R' - R + m1a1 + m2a2 + m3a3

Where mi are integers.  Such a lattice of building blocks is known as Bravais lattice. Crystal structure could be understood by combination of properties of building block (basis) and of Bravais lattice. The natural way to explain crystal structure is the set of point group operations that involve operations applied around the point of lattice.

One-Dimensional Lattices - Chains:

19_One dimensional lattices.jpg

White and black circles are atoms of different type. a is primitive lattice with one atom in primitive cell; b and c are composite lattice with two atoms in cell.

Two-Dimensional Lattices:

There are 5 basic classes of 2D lattices

2026_five classes of 2-dimensional lattices.jpg

The Reciprocal Lattice:

Crystal periodicity leads to several significant consequences.  Namely, all properties, say electrostatic potential V, are periodic

V(r) - V(r + an), an = n1a1 + n2a2 + n3a3

It implies Fourier transform.  Generally oblique co-ordinate system is introduced, axes being directed along ai.

aibk - 2πδi,k

Vectors bk are known as basic vectors of reciprocal lattice. As a result, one can create reciprocal lattice using those vectors, elementary cell volume being (b1[b2,b3]) =(2π)3/V0.

Modern research in solid state physics:

Present research topics in solid state physics comprise:

  • Strongly correlated materials
  • Quasicrystals
  • Spin glass
  • High-temperature superconductivity

Quasicrystals:

Essential condition for having crystalline phase is lattice periodicity and crystal cannot have five-fold rotational symmetry. It will be fairly interesting to mention that there are physical systems with several properties of usual crystalline state but without three dimensional lattice periodicity like Al0.86, Mn0.14 Al6 CuLi3. Ga0.2.1 Mg038 Zn10.41 and CU0.2 Al0.65 Fe0.15 etc. These are known as quasicrystals, systems with long range order but with five fold symmetry axes in shape of icosahedron, that exclude lattice periodicity. A function is termed to be quasiperiodic if it can be termed as sum of two periodic structures whose periods are incommensurate. The ideal quasi-crystal will, thus, be any structure which is made up of infinite repetition in space of two or more distinct unit cells and that show long range quasiperiodic translations and long range orientational order. Because of the presence of different unit cells, quasicrystals are differentiated by existence of two or more incommensurate length scales.

Superconductivity:

Superconductivity is the property shown by certain materials at very low temperatures.  Materials found to have this property comprise metals and their alloys (tin, aluminum, and others), few semiconductors, and certain ceramics called as cuprates which have copper and oxygen atoms.  The superconductor conducts electricity without resistance, unique property.  It also repels magnetic fields completely in the phenomenon called as Meissner effect, losing any internal magnetic field it might have had before being cooled to the critical temperature.  Due to this effect, few can be made to float continually above strong magnetic field.

Spin Glass:

Another class of order which may take place in magnetic materials at low temperatures is spin glass. Name is meant to recommend frozen in (long-range) disorder. Experimentally onset of spin glass is signaled by the cusp in magnetic susceptibility at T1 (freezing temperature) in zero magnetic field. Below T1 there is no long-range order. Classic examples of spin glasses are dilute alloys of iron in gold (Au:Fe, also Cu:Mn, Ag:Mn, Au:Mn and many other examples). Critical ingredients of spin glass appear to be (a) competition among interactions as to preferred direction of spin (frustration), and (b) randomness in interaction between sites (disorder).  Study of spin glasses fall in broad category of study of disordered systems, comprising random field systems (such as diluted antiferromagnets), glasses, neural networks, and optimization and decision problems.

Strongly correlated materials::

Strongly correlated materials are the wide class of electronic materials which show unusual electronic and magnetic properties, like metal-insulator transitions or half-metallicity. The necessary characteristic which states these materials is that behavior of their electrons can't be explained effectively in terms of non-interacting entities. Theoretical models of electronic structure of strongly correlated materials should comprise electronic correlation to be precise.

Strongly correlated materials have partly filled d- or f-electron shells having narrow energy bands. One can no longer consider any electron in material as being in sea of averaged motion of others (also called as mean field theory). Every single electron has complex influence on neighbors.

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