Institute of Solid State Physics Technische Universität Graz

5. Tight Binding / Graphene

Oct 15, 2018

Tight binding, one atomic orbital

 cHaa MOa  cH ma MOmexp( ihkajkalka (123 )) small terms nearest neighbors m

Ecka a a small terms  For only one atomic orbital in the sum over valence orbitals

   Ecka a a cH a a MOa  cH a a MOmexp( ihka ( 123jka lka )) nearest neighbors m

one atomic orbital

  ik m Etek   m      aMOa rH r trHraMOam     Tight binding, simple cubic

  Ete  ik m m ik a ik a Eteeeeee  ikxx a  ik a yy  ikzz a  ik a 

 2cos()cos()cos()tkakakaxyz  

22 Effective mass m*  dE2 2ta2 dk 2 Narrow bands  high effective mass Density of states (simple cubic)

Calculate the energy for every allowed k in the Brillouin zone

Etkakaka 2 cos(xyz )  cos( )  cos( ) http://lamp.tu-graz.ac.at/~hadley/ss1/bands/tbtable/tbtable.html Etkakaka 2 cos(xyz )  cos( )  cos( ) Tight binding, fcc

  Ete  ik m m Density of states (fcc)

Calculate the energy for every allowed k in the Brillouin zone

http://lamp.tu-graz.ac.at/~hadley/ss1/bands/tbtable/tbtable.html Tight binding, fcc

Christian Gruber, 2008 Tight binding, fcc

http://www.phys.ufl.edu/fermisurface/ Table of tight-binding calculations

http://lampx.tugraz.at/~hadley/ss1/bands/tbtable/tbtable.html Graphene

C1 C2   31 C2 a C1c1 C2 aaxa1 ˆˆy 1 22 C1c1 C2c2 C1  31  aaxa2 ˆˆy C2 a 2 C1c1 C2 22 C1 C2

Two atoms per unit cell

Graphene has an unusual dispersion relation in the vicinity of the Fermi energy.  odel is ala j

  orbitals kk z

p  HE alac r j  zz

      Substitute this wave thisfunction into the Schrödinger equation Substitute 1 2 12 12   ka lka c r j  b i

  2 carbon atoms / unit cell exp For graphene, are the valence orbitals , jl a    The standard guess for the wave The standard guess for function in the tight binding m kapabpb

 2 carbon atoms / unit cell

     exp i jka lka c r jalac rjala kapabpb  1 2 zz 12 12 jl,

HE kk   Multiply by  *  r  and integrate paz

the orbital for the atom at j = 0, l = 0.

  ik m cHaa a cH ba b e small terms m   ik  m  Ecaaa c bab e small terms   m

1 0 m sums over the nearest neighbors 2 carbon atoms / unit cell

To get a second equation for ca and cb

 HE    * r Multiply kk by pb z   and integrate

the orbital for the atom at j = 0, l = 0.

  ik m cHab a  e cH bb b small terms m

  ik  m Ecaba e c bbb + small terms  m 0 1 Write as a matrix equation Tight binding graphene

   ik  m aaHE  ab H e m c a    0  ik m c baHe bb HE b m

m sums over the nearest neighbors.

There will be two eigen energies for every k.

N orbitals / unit cell results in N bands

  aaH

tH  ab Tight binding graphene

  ik  m  Ete  m    0 te ik m E m 

   33kakayy ka ka eiik  m 1expxx   exp i   22 22 m      ka 1 ka 2  a  31 1 aaxaˆˆy 1 22   31 aaxa2 ˆˆy a 2 22 unit cell

There will be two eigen energies for every k. Solve for the dispersion relation

 33kakayy ka ka  Etii 1expxx exp 22 22   0  33kakayy ka ka ti1exp xx   exp  i   E 22 22 

33kaka ka ka 1expiixxyy exp  22 22    2 2 333kakayyy  ka ka ka ka  Etexp ixxx   1 exp i   i   0    22  22 22   ka ka ka 33kaxxy ka yy3kax expii   exp   i 1 22 2222  Graphene dispersion relation

2 33kaka ka ka  Et2 3 2 cosxxyy 2 cos 2 cos ka 0 y 22 22

cosab cos a cos b  sin a sin b cosab cos a cos b  sin a sin b cos 2aa 2cos2  1

2 3ka ka  ka  Et2214cos x cosyy  4cos  0   22  2

3ka ka  ka Et14cosx cosyy 4cos2     22  2 Tight binding, graphene

3ka ka  ka Et14cosx cosyy 4cos2     22  2

K  M

www.physics.umd.edu/courses/Phys732/hdrew/spring07/ Schoenenberger%20tutorial%20on%20CNT%20bands.pdf K  M

2.8

Another band is included here. http://onlinelibrary.wiley.com/doi/10.1002/adfm.201101241/full http://lamp.tu-graz.ac.at/~hadley/ss1/bands/tbtable/dispgraphene.html Graphene

Mobility: 200000 cm2 /V s suspended, ~20000 cm2 /V s otherwise Carbon nanotubes - rolled up graphene

armchair

zig-zag

chiral

www.physics.umd.edu/courses/Phys732/hdrew/spring07/ Schoenenberger%20tutorial%20on%20CNT%20bands.pdf (m,n) notation

http://www.personal.rdg.ac.uk/~scsharip/tubes.htm Carbon nanotubes

3ka ka  ka Et14cosx cosyy 4cos2     22  2

metallic (5,5) armchair tube http://lamp.tu-graz.ac.at/~hadley/ss1/bands/tbtable/CNTs.html Carbon nanotubes

metallic m - n = 3Z semiconducting

www.physics.umd.edu/courses/Phys732/hdrew/spring07/ Schoenenberger%20tutorial%20on%20CNT%20bands.pdf Table of tight-binding calculations

http://lampx.tugraz.at/~hadley/ss1/bands/tbtable/tbtable.html Bloch waves in one dimension

http://lampx.tugraz.at/~hadley/ss1/bloch/bloch.php Band Structure in one dimension

http://lampx.tugraz.at/~hadley/ss1/bloch/bloch.php