SUBLATTICESSUBLATTICES&&

‘‘SUBCRYSTALS’SUBCRYSTALS’

MATERIALS SCIENCEMATERIALS SCIENCE&&

ENGINEERING ENGINEERING

Anandh Subramaniam & Kantesh Balani

Materials Science and Engineering (MSE)

Indian Institute of Technology, Kanpur- 208016

Email: anandh@iitk.ac.in, URL: home.iitk.ac.in/~anandh

AN INTRODUCTORY E-BOOKAN INTRODUCTORY E-BOOK

Part of

http://home.iitk.ac.in/~anandh/E-book.htmhttp://home.iitk.ac.in/~anandh/E-book.htm

A Learner’s GuideA Learner’s GuideA Learner’s GuideA Learner’s Guide

The concept of sublattices (and a new concept of subcrystals based on this) are useful in understanding ordered structures.

The use of the term superlattice* implies that it is composed of more than one sublattice. Typically all sublattices are identical, but with the origin of one shifted w.r.t to the other. Populating a sublattice with a species/motif (‘a sub-motif?**’) gives us a ‘subcrystal’. Subcrystals may be identical (same species sits in both the subcrystals) or may be

different (‘sib-motif’ populating the sublattices may be different). Subcrystals combine (interpenetrate) to give a supercrystal (analogous to the superlattice)

Sublattices and Subcrystals

* Usually the use of the prefix ‘super’ implies an highly enhanced property, like in superconductivity, superfluidity, superparamagnetism etc. In the case of the superlattice it just implies that it is made of more than one ‘sublattice’

** Sub-motif may be thought of as a part of the motif of the supercrystal.

These concepts will become clear on considering examples

Click here to see connection between superlattices and ordered structuresClick here to see connection between superlattices and ordered structures

Concept of SublatticeLet us revisit the crystal (X) made of up arrows and down arrows to understand the concept of sublattices

This crystal can be understood as a superposition of two crystals as below

+

SX2

SX1

X

X + SX2SX1=

Sub-Crystal-1 (SX1) consists of only up arrows and Sub-crystal-2 (SX2) consists only of down arrowsThe crystal can be called a ‘Super-Crystal’ (supercrystal)

Example-1

‘Super-Crystal’ (X)

Sub-Crystal-1 (SX1)

Sub-Crystal-2 (SX2)

Correspondingly we can think of a ‘Superlattice’ (L)

Which can be broken into two Sublattices → two interpenetrating sublattices

+

SL2

SL1

L

L + SL2SL1=

Sub-Lattice-1 (SL1) and Sub-Lattice-2 (SL2) combine to create the lattice (L)

SubLattice-1 (SL1)

SubLattice-2 (SL2)

Lattice

If the lattice parameter of the crystal is ‘a’

then Sublattice-1 (SL1) is displaced with respect to Sublattice-2 (SL2) by a/2

Note that in the crystal SL2 (or equivalently SL1) is not a set of lattice points

Let us consider another example to understand the concept of sublattice (now in 2D)

This is the familiar crystal which we had considered before

X

Example-2

‘Super-Crystal’ (X)

Square Crystal

Let us analyze this crystal in terms of subcrystals and sublattices

SX2

SX1

X + SX2SX1=

Sub-Crystal-1 (SX1) consists of only green circles and Sub-crystal-2 (SX2) consists only of brown

‘Super-Crystal’ (X)

SL2

SL1

L + SL2SL1=

Sub-Lattice-1 (SL1) and Sub-Lattice-2 (SL2) combine to create the lattice (L)

L

Note that in the crystal SL2 (or equivalently SL1) is not a set of lattice points

Example-3

Let us consider a 3D example of a Supercrystal (superlattice)

If the brown spheres are Na+ ions and white spheres are Cl ions (of different sizes) this can be thought of as a

model for NaCl

This crystal can be thought of a two interpenetrating

subcrystals:

SX1 = FCC SL1 decorated by white metallic balls

SX2 = FCC SL2 decorated by brown metallic balls

NaCl

X+

SX2SX1 =‘Super-Crystal’ (X)