Brillouin Zone of a 2D Square Lattice: Tight Binding Approximation

CaptionResourceCreateNotebook

This Demonstration considers the Brillouin zone (BZ) s-band dispersion relations for a two-dimensional square crystal lattice under the tight binding approximation. Plots are shown for the first energy s-band dispersion in the first and extended Brillouin zones. You can compare the parabolic dispersion curves of free electrons in free space and the linear dispersion curves of Dirac-electrons (as in graphene at K-points).

Initialization CodeResourceCreateNotebook

If you provide initialization code, include a SaveDefinitions->True option in the Manipulate.

ManipulateResourceCreateNotebook

Manipulate[
If[how2D,ContourPlot,DensityPlot,Plot3D]@@{{##,PlotPoints->5},{##,PlotPoints->15},{##,PlotPoints->20},{##,ImageSize->{400,400}}}[[quality]]&[
 If[SqrOrLin,(x^2)+(y^2),Abs[(Sqrt[(x^2)+(y^2)])],-2(Cos[π x]+Cos[π y])],{x,-extendBZ,extendBZ},{y,-extendBZ,extendBZ},If[how2D,Contours->20,PlotRange->All,Contours->5],
BoxRatios->If[how2D,Automatic,Automatic,{1,1,1}],If[how2D,Frame->box,Frame->box,Boxed->box],RegionFunction->cut3D,Mesh->mesh,MeshStyle->Directive[Opacity[0.8],Gray],ImageSize->{380,380},BoundaryStyle->{Thick,Gray},ColorFunction->cf,Axes->axes,
AxesLabel->{(Subscript[k, x])[π/a],(Subscript[k, y])[π/a],Row[{(Subscript[Style["E",Italic], k]),"(eV)"}]},LabelStyle->Directive[Bold,13,Gray],If[how2D,PlotRange->All,PlotRange->All,ViewPoint->vp],MeshFunctions->{#3 &},ImagePadding->{{25,45},{25,45}}],
{{extendBZ,1,"extend BZ"},1,4,1,ControlType->Setter},Delimiter,
Grid[{{  Column[{
Control[{{mesh,False,"show mesh"},{False,40},ControlType->Checkbox}], 
Control[{{box,True,"  show box"},{False,True},ControlType->Checkbox}]    }],"",
Column[{
Control[{{axes,False,"           show axes"},{False,True},ControlType->Checkbox}],
Control[{{cut3D,True,"cut segment in BZ"},{True,If[how2D,True,True,Function[{x,y},x<0||y>0]]},ControlType->Checkbox}]
   }] }}],
{{quality,2,"quality"},{1->"low",2->"medium",3->"high"}}, {{cf,"Rainbow","colors"},{Automatic->"Automatic","Rainbow"->"Rainbow","DarkRainbow"->"DarkRainbow",Hue->"Hue","CMYKColors"->"CMYKColors","ThermometerColors"->"ThermometerColors","TemperatureMap"->"TemperatureMap","LightTemperatureMap"->"LightTemperatureMap","SouthwestColors"->"SouthwestColors","BlueGreenYellow"->"BlueGreenYellow","LightTerrain"->"LightTerrain","SandyTerrain"->"SandyTerrain","IslandColors"->"IslandColors","Pastel"->"Pastel","PearlColors"->"PearlColors","RedBlueTones"->"RedBlueTones","LakeColors"->"LakeColors","FallColors"->"FallColors","RoseColors"->"RoseColors","WatermelonColors"->"WatermelonColors","GrayYellowTones"->"GrayYellowTones","GreenBrownTerrain"->"GreenBrownTerrain","GrayTones"->"GrayTones"}},Delimiter,
 {{how2D,True,"plot "},{True->"contour",False->"density",Plot3D->"pseudo3D"},ControlType->SetterBar},
{{vp,{1.3,-2.4,2},"3D view"},{{1.3,-2.4,2}->"default",Above->"above",Front->"front"},Enabled:>how2D==Plot3D},Delimiter,
Style["for comparison:",Bold],
 {{SqrOrLin,None,""},{None->"lattice",True->"square",False->"linear"},ControlType->SetterBar},
Style["lattice -> lattice electron dispersion",Bold,Gray],
Style["square-> free electron dispersion",Bold,Gray],
Style["linear -> Dirac-electron dispersion",Bold,Gray],ControlPlacement->Left,TrackedSymbols->True]

SnapshotsResourceCreateNotebook

Source & Additional Information

Contributed ByResourceCreateNotebook

Vladimir Gavryushin

Details & CitationsResourceCreateNotebook

The energy structure of crystals depends on the interactions between orbitals in the lattice. The tight binding approximation (TB) neglects interactions between atoms separated by large distances, an approximation which greatly simplifies the analysis. In solid-state physics, the TB model calculates the electronic band structure using an approximate set of wave functions based upon superposition of orbitals located at each individual atomic site.

In a TB approximation including only first nearest neighbor s-orbital, the band structure, i.e. the electron energy dispersion in the Brillouin zone of the crystal, is given by

(Subscript[E, (Overscript[k, →])])=(Subscript[E, s])+(Underscript[Σ, (Overscript[k, →])])(e^(i(Overscript[k, →])(Overscript[k, →])))ssσ((Overscript[R, →])) (1)

where ssσ is the overlap integral between s-orbitals, R is the translation vector of the lattice, and ħk is the "crystal momentum"—the quantum number for periodic systems. The band structure for a simple cubic lattice can now be readily calculated. Assuming that the bond integrals couple only to the first four nearest neighbors with position vectors R equal to (±a,0) and (0,±a), (1) gives

(Subscript[E, (Overscript[k, →])])=(Subscript[E, s])+2 s s σ(cos((Subscript[k, x])a)+cos((Subscript[k, y])a)).

(a)

(b)

(c)

Figure: (a) lattice fragment; (b) four first nearest neighbors included in transfer; (c) the first Brillouin zone.

Setter bars let you compare the lattice electron behavior with the parabolic dispersion curves of the free electrons in free space,

(Subscript[E, k])=((ℏ^2)/(2(Subscript[m, e])))(((Subscript[k, x])^2)+((Subscript[k, y])^2)),

and with the linear dispersion curves of Dirac electrons (as in graphene K-points),

(Subscript[E, k])=(Subscript[v, g])ℏ(Sqrt[((Subscript[k, x])^2)+((Subscript[k, y])^2)]).

Snapshot 1: constant energy contours for the s-band in the first Brillouin zone of a 2D one-base simple square lattice

Snapshot 2: constant energy contours for the free Dirac electrons in free space

Snapshot 3: constant energy contours for the 2D square lattice in an extended 3×3 Brillouin zone

Snapshot 4: energy dispersion for the s-band in an extended 3×3 Brillouin zone of a 2D square lattice

Snapshots 5 and 6: same energy dispersion viewed from top and front

References:

D. G. Pettifor, Bonding and Structure of Molecules and Solids, Oxford: Clarendon Press, 1995.

C. Kittel, Solid State Physics, Hoboken, NJ: John Wiley & Sons, Inc., 1996.

ReferencesResourceCreateNotebook

References for demonstration.

Submission NotesResourceCreateNotebook

Additional information for the reviewer.

KeywordsResourceCreateNotebook

solids

bonding

band structure

tight binding

simple cubic lattice

energy structure

Brillouin zone

energy dispersion

CategoriesResourceCreateNotebook

Related DemonstrationsResourceCreateNotebook

Link to a related demonstration

External LinksResourceCreateNotebook

Brillouin Zone Sampling of a Periodic Chain with N Sites

Brillouin Zone Sampling of Chain with N Sites and Modified Periodic Boundary Conditions

Brillouin Zone of a Single-Walled Carbon Nanotube

Basic Structure, Plane Orientation, and Facetting in the Face-Centered and Body-Centered Cubic Crystal Structures

Crystal Viewer

LatticeData

Cubic Lattice

Related SymbolsResourceCreateNotebook

SymbolName (documented Wolfram Language symbol)

Published DateResourceCreateNotebook

2011-03-07

CompatibilityResourceCreateNotebook

AR SupportResourceCreateNotebook

Wolfram Language VersionResourceCreateNotebook

6

Cloud SupportResourceCreateNotebook