Closing the band gap of a 2D semiconductor with an electric field
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BAND Electric field Vector field Band gap Semiconductor

This tutorial focuses on a monolayer of Molybdenum disulphide MoS_2,
because this material is of interest for transistors, in particular
the switching to and from a conducting state. It is based on a CECAM
workshop exercise.


Create the MoS_2 monolayer
==========================

You can use the MoS_2 monolayer from our database, follow step 1 of
the MoS2 response tutorial.


Analyze the DOS and band structure
==================================

   **1.** Select **Relativity (ZORA)** → **Scalar** in the Main menu
   **2.** Tick the **Calculate PDOS** box
   **3.** Tick the **Calculate band structure** box
   **4.** Select **Details** → **K-Space Integration** in the panel bar
   **5.** Set **K-space grid type** to "Symmetric"
   **6.** Run the calculation (**File** → **Run**)
 [image][image]

Note:

  For some materials, like the MoS_2 monolayer in this tutorial, the
  high symmetry k-points are important. For those materials, it is
  important to sample the k-space symmetrically.

After the calculation is finished, the band structure can be
visualized.

   **4.** Open the band structure by selecting **SCM** → **Band Structure** in the menu bar

The bands are colored according to the main contribution, also known
as ‘fat band analysis’. The band structure will look as follows:

[image]

To receive band structure curves that are more smooth, the delta-K
parameter of the interpolation can be adjusted.

   **5.** Perform a new calculation with the same settings as above
   **6.** Select **Properties** → **Band Structure** in the panel bar
   **7.** In the Interpolation delta-K enter "0.03"
   **8.** Run the calculation and open the **Band Structure** again
 [image]

The bands should look much more smooth now. If you zoom in and hover
over the bands you should see an indirect band gap from the gamma
point to the K point, which is on the edge of the Brillouin zone. (See
the indirect band gap at the horizontal black line.)


Electron-hole transport
=======================

For semiconductors the shape of the bands near the Fermi level tells
you something about how easy the electrons (in the bottom of the
conduction band) or holes (in top of the valence band) can move. The
inverse of the curvature is called the effective mass of the electron
or hole. A small curvature means a large effective mass of the
electron or hole, and a large curvature means a large effective mass.
To calculate this effective mass:

   Perform a new calculation with the same settings as before
   Tick the Effective mass box in **Properties**  → **Effective Mass** in the panel bar
   Run the calculation. The results will be visible in the Output (.out file)

Tip:

  See also the BAND tutorial Band Structure and Effective Mass Tensors
  of Phosphorene


Fix the band gap
================

The band gap (as read from the band structure plot) is about 1.7 eV,
whereas experimentally it is 1.9 eV (see e.g. Heinz et al. [1] ) We
have used the LDA functional, which is known to underestimate the gap.
You could try to improve the band gap by using a model potential. For
systems that are not crystals, the most convenient choice is the GLLB-
SC functional. (XC functional: Model → GLLB-SC) Now the Quasiparticle
gap should be a bit higher, since the GLLB-SC includes explicitly the
so-called derivative discontinuity. The SCF convergence can be more
difficult with GLLB-SC. We leave it as an exercise to find good
settings for GLLB-SC. Consider increasing the basis set, modifying the
k-point grid or changing the SCF Method in any of the Details panels.

Note:

  The band gap in the log file can be different from the one obtained
  from the band structure. This is because the k-points are different
  in the initial calculation and the subsequent band structure
  calculation.


Applying an electric field
==========================

An electric field will reduce the gap significantly. You will need
quite strong fields to make the material conducting.

   **1.** Switch back to the LDA functional
   **2.** In the panel bar, select **Model** → **Electric field**
   **3.** In the third **Electric field** field (along **z**) enter "3.0" V/Å.
   **4.** Run the calculation and open the **Band Structure**
 [image]

You will find the band character can change a bit at higher fields and
you could determine through multiple calculations at what field
strength the material will become conducting.


Analyzing the charge
====================

The charge can be analyzed after you applied an electric field.

   Select **SCM** → **View** in the menu bar
   Select all atoms and select **Add** → **Plane Colored** in the menu bar
   Select at the right **Fit density** → **Fitted deformation density**
   Select the atoms with right click and select **Atom info** → **CM5 Charges** to show the charges

At a field of 3 Volt/Angstrom and some tweaking (You can change these
features by opening the details by clicking the arrow in front of
**Plane: Colored**)

[image]

Here you can see that the charges on the two different S atoms are
different because of the electric field.


Improving the accuracy
======================

Many aspects influence the results of a calculation. Some are
technical, such as the choice of basis set, k-points and integration
accuracy; other are theoretical, such as the choice of functional. You
can also consider computationally more expensive spin-orbit coupling
instead of scalar relativistic. The relativistic effect is small for
light atoms and grows with the charge of the nucleus.

The choice of XC functional is not so straightforward. However, to use
a GGA is generally better than using the plain LDA. Among the GGAs the
PBE functional is a reasonable choice or more modern metaGGAs such as
MN15L and SCAN. Finally, the DZ basis set is usually too small, and
one should preferably use a TZP one. For the gap (especially when
p-electrons are involved) also the spin-orbit might be needed.


References
==========

[1] K. Mak, C. Lee, J. Hone, J. Shan, T.F. Heinz *Atomically Thin
    MoS2: A New Direct-Gap Semiconductor*, Physical Review Letters
    105, 136805 (2010)
