Why the band structure of a slab is not smooth?

Question

I am trying to calculate the band structure of a TiO2-Rutile slab, but the result I got is not very smooth, and the result shows it is an indirect gap material, but the bulk phase of TiO2-Rutile is a direct gap material. My question is:

1. Is it possible that the energy gap of the material is direct in the bulk but indirect in the slab?
2. Is it normal that the band structure of a slab is not smooth? I can see the result of Ni 100 surface bandstructure on the vasp tutorial, it seems also not very smooth.

Here are the INCAR,POSCAR and KPOINTS files I use:

System=TiO2 surface DOS
#Startparameter for this run:
    ISTART =1         #job   :0-new  1-continue 2-samecut   charge:0-initial
    ICHARG =11         #charge:0-initial orbitals 1-from CHGCAR 2-ISTART=0 superposition of atoms

#Electronic Properties
   ISMEAR =1         #part.occupancies -5-Blochl -4-tetra -1-fermi 0-guas;when calculate band,don't use ISMEAR=-5  conductor-1  semiconductor and insulator-0
   SIGMA  =0.05      #broadening in eV, when calculate Band structure, ISMEAR, SIGMA use default values   
   ISPIN  =2         #spin polarized calculation? 1-no 2-yes default-2
   LORBIT = 11
#Electronic Relaxation
    PREC   =Accurate
    ENCUT  =520       #energy cutoff in eV: Default-largest ENMAX from POTCAR-file
    EDIFF  =0.1E-04   #stopping-criterion for ELM
    NELM   =100       #the maximum number of electronic SC steps,default=60
    NELMIN =3         #the minimum number of electronic SC steps,default=2
#    NELMDL =3        #number of non-selfconsistent steps at the begining，default=5-10

#DOS
    NEDOS  =2000
    EMIN   =-10
    EMAX   =10

#Write file or not?
#    LWAVE  =.FALSE.   #write WAVECAR,default= .TRUE. 
#    LCHARG =.FALSE.   #write CHG/CHGCAR,default= .TRUE. 

#Ionic relaxation
#    EDIFFG =0.1E-03   #stopping-criterion for IOM, default: EDIFFG = EDIFF×10 
#    NSW    =200        #number of steps for IOM.  0-default
#    IBRION =2         #ionic relax: ionic relaxation: 0-MD 1-quasi-New 2-CG
#    ISIF   =2         #stress and relaxation, when IBRION=0, ISIF=0,other default values is 2.
#   POTIM  =0.10      #time-step for inoic-motion
#   LREAL  =.FALSE.   #.TRUE.=true space  .FALSE.=reciprocal space = default.

#Parallelization flags
    NCORE=8
    #KPAR=6

TiO2_mp-2657_computed\(1\1\0)
1.0
        2.9691998959         0.0000000000         0.0000000000
        0.0000000000         6.5806999207         0.0000000000
        0.0000000000         0.0000000000        40.7333984375
   Ti    O
   10   20
Direct
     0.500000000         0.000000000         0.129769996
     0.500000000         0.500000000         0.210549995
     0.500000000         0.000000000         0.291330010
     0.500000000         0.500000000         0.372099996
     0.500000000         0.000000000         0.452879995
     0.000000000         0.500000000         0.129769996
     0.000000000         0.000000000         0.210549995
     0.000000000         0.500000000         0.291330010
     0.000000000         0.000000000         0.372099996
     0.000000000         0.500000000         0.452879995
     0.500000000         0.304580003         0.129769996
     0.500000000         0.804579973         0.210549995
     0.500000000         0.304580003         0.291330010
     0.500000000         0.804579973         0.372099996
     0.500000000         0.304580003         0.452879995
     0.500000000         0.695420027         0.129769996
     0.500000000         0.195419997         0.210549995
     0.500000000         0.695420027         0.291330010
     0.500000000         0.195419997         0.372099996
     0.500000000         0.695420027         0.452879995
     0.000000000         0.000000000         0.098200001
     0.000000000         0.500000000         0.178979993
     0.000000000         0.000000000         0.259759992
     0.000000000         0.500000000         0.340530008
     0.000000000         0.000000000         0.421310008
     0.000000000         0.000000000         0.161339998
     0.000000000         0.500000000         0.242119998
     0.000000000         0.000000000         0.322899997
     0.000000000         0.500000000         0.403679997
     0.000000000         0.000000000         0.484450012

Special k-points for band structure
10  ! intersections 
line-mode
reciprocal
    0.0000000000     0.0000000000     0.0000000000 1    GAMMA
    0.5000000000     0.0000000000     0.0000000000 1    X 


    0.5000000000     0.0000000000     0.0000000000 1    X 
    0.5000000000     0.5000000000     0.0000000000 1    S 


    0.5000000000     0.5000000000     0.0000000000 1    S 
    0.0000000000     0.5000000000     0.0000000000 1    Y 


    0.0000000000     0.5000000000     0.0000000000 1    Y 
    0.0000000000     0.0000000000     0.0000000000 1    GAMMA

Here is the result I got:

OUTCAR ->
NSPIN  =    2; NKPTS  =   40;
NBANDS =  150; Efermi =  -2.8040
------------------------------------------------------
                 SPIN_UP               SPIN_DN
------------------------------------------------------
      IND          111                   111
 CBM  ENG        -1.63250              -1.63220
      KPT  0.5000 0.5000 0.0000  0.5000 0.5000 0.0000
          ____________________________________________
      IND          110                   110
 VBM  ENG        -2.81420              -2.83650
      KPT  0.1667 0.5000 0.0000  0.1111 0.0000 0.0000
          ____________________________________________
   GAP           1.18170               1.20430
               inDirect_Gap          inDirect_Gap
                            1.18170
                          inDirect_Gap

An update after a new test, if I use 40 intersections between two special points. The band structure looks better obviously.

Answer 1

You should take Binh Tien's advice for the continuity part of your problem. It has also been answered somewhat here:

How to ensure a smooth band structure?

As for the gap issue, adding to Phil Hasnip's answer, a similar question has also had several proposals for answers here :

How to generate the high symmetry paths for band structure calculations?

These might be enough for you to fix your issue.

Allow me to stress again on the fact that, "The 2D slab does not have the same symmetry as the full 3D slab, and there is no requirement for the conduction band minima and valence band minima to be at the same points in reciprocal-space"

Don't forget to give that green checkmark for the answers that help you resolve your issue and to comeback and tell us how you did it yourself so that we can close out this issue !

Answer 2

1. Is it possible that the energy gap of the material is direct in the bulk but indirect in the slab?

Yes, this is possible (as is the reverse). The 2D slab does not have the same symmetry as the full 3D slab, and there is no requirement for the conduction band minima and valence band minima to be at the same points in reciprocal-space. As you make the slab thicker and thicker, you would usually expect to recover the properties of the 3D bulk materials, although strictly speaking the symmetry is not fully recovered whilst it is only 2D-periodic.

2. Is it normal that the band structure of a slab is not smooth?

No, this is not physically sensible. The curvature of the band in reciprocal-space is related to the particle's effective mass, $m$, by

\begin{align} \left[m^{-1}\right]_{ij} = \frac{1}{\hbar^2} \frac{\partial^2 E}{\partial k_i \partial k_j}. \end{align}

Except for a fully localised band, the effective mass should be finite. The jagged, non-differentiable parts of your band-structure are artefacts of the calculation or plotting. This is probably due to the finite sampling of k-points (10 is not very many for such a small real-space cell), and I expect they would disappear if you used more intermediate points or a larger real-space cell. Note that your cell has an in-plane aspect ratio of approximately 1:2 so the k-points are spaced twice as far apart in one direction, and this is where the bands appear most jagged.

Jagged band-structures can also be artefacts of the plotting software, which usually joins bands together based simply on their energy, rather than the corresponding physical state. Using additional information, such as band-gradients or projections, can allow a more physical matching of states across the Brillouin zone, and improve the quality of the plot.