Script for calculating the effective mass in semi-conductors.

[Quantamania]

I have read the papers describing the effective mass calculations posted in this thread.  However, they are for three-dimensional crystals.

My research, which was recently approved by my proposal committee last Wednesday, revolves around pi-stacking and modeling their electron transport properties.  I have made some interesting findings regarding their band structures, how they behave under phonon movement through them, and how their properties change from small molecular clusters to bulk pi-stack periodic structures.

Pi-stack structures are remarkably simple one-dimensional conducting wires.  Based on their band structures, three k-points in one layer are the same, while another three k-points are also equal in a separate layer.  These layers, I define as KGM and HAL, have different periodicity in Bloch wavefunctions.  This difference produces the energy changes seen in their band structures (the example I attached is for benzene at 4 angstroms apart).

The picture below is the pi-stack crystal model I am using, a hexagonal crystal.


How can the equations for conductivity, effective masses, and other properties in the script be adapted for one-dimensional conducting wires?  I tried to apply the script to benzene pi-stacks, but got errors that prevented me from getting the numbers.

Can you please explain to me the changes that I would need to make in order to use the equations?  The band structure of a pi-stack shows inflection at anytime crystal momentum moves from the KGM layer to the HAL layer or vice versa.  The bandwidth inside the layers is essentially zero, and distance between rings controls the bandwidth between k-point layers.  This gives us a conducting system with both a direct band gap and indirect band gap, each represented by intramolecular and intermolecular conduction.  The direct band gaps in the KGM and HAL layers are quite insensitive to inter-ring distance, but the indirect band gap is especially sensitive to such effects, so its properties are highly tunable in this respect.

This would really help with my research as it currently stands.

[Nordland]

I am properly wrong - but I don't think there is much to gain from investigating the effective mass in this kind of system. The effective mass approximation assumes, that the bands are parabolic around a minimum energy. This is the case for most three dimension crystals, but it can also be true for 1d or 2d systems.

Since your energy bands are technical flat with very little sign of parabolic nature, I think I would go straight ahead and use the standard NEGF transport calculation.

[Nordland]

On a note, I should say, that I could generalized the script to be able to deal with 1d structures, and it would properly be pretty good for determine the bands gaps, but the effective mass in your system will still be a fishy

[Quantamania]

I have previously tested the NEGF methods on these pi-stacks and junctions built from them, but have yet to see if known equations for effective mass can work on them.

The flatness of the pi-stack bands arises from the degeneracy of k-points, an effect imposed by the large dimensions of the rhombi making up the faces of the crystal.  However, they are not flat across K-H, Gamma-A, or M-L.  At the midpoints of these lines is the inflection point, where the slope of the band changes.  Based on Roald Hoffmann's intuitive analysis of solid-state physics, the bands I have are behaving the way predicted by purely p-orbital arrangements.  They start at a high energy, then lose energy as phase of adjacent orbitals is flipped in a continuous manner.

Initially, at K, the slope of the band is zero.  At P (the midpoint of K-H), this slope is the greatest (most negative).  It is also where the second derivative becomes zero.  The slope begins to increase back up to zero as it goes to H once it passes P.  The same thing holds for Delta (between Gamma and A) and the midpoint between M and L (do you know the symbol for it?).  So I get three bandwidth regions if I use the default k-point set (G-M-L-A-G-K-H-A), since I have M-L, A-G, and K-H in that pathway.  The example I posted in the last post was done using the same k-point pathway as the graphite calculations I did earlier.  I did notice the non-parabolic behavior of these bands.  What should we do in these instances?

[Anders Blom]

The M-L path is called U.

I don't think the concept of effective mass applies in this case. It's a quantity used in simpler models, where the band structure is close to parabolic, such that you don't need the detailed band structure, but it suffices to use a simplified picture of it, i.e. a perfect parabola. If the band structure doesn't look like that, there is no point in trying to compute an effective mass, since it cannot be used for anything meaningful...

[Quantamania]

Thanks for the clarification.  The bands that I generally get tend to be non-parabolic, but rather sigmoid.  However, if we look at the segments, K-P and P-H, they resemble parabolas.  The same thing holds for M-U, U-L, Gamma-Delta, and Delta-A segments of the band structure.  What should we do in this situation?  Thus, it seems that the useful information we have include these right now:

Bandwidth of valence and conduction bands between k-point layers (K to H, Gamma to A, M to L)
Direct band gaps (between valence and conduction bands within KGM and HAL layers)
Indirect band gap (highly tunable with inter-ring distance, represented by Delta, U, and P lines of the Brillouin zone)
General slope behavior of valence and conduction bands (do they run in opposite directions or the same directions?)

Can you think of any other properties that can be gleaned from the simple pi-stack model?  If so, I would like to consider them for my dissertation work.

[Quantamania]

I have some new findings regarding my pi-stack structures...they are quite elastic.  That means they have small energy barriers against phonon movement, but have stable structures consisting of equal inter-ring distances.

The method I used to deduct this was to perform calculations using GAUSSIAN03 with the same models I am discussing here.  As before, I have a periodic box for the arenes, with room for two molecules.  This supercell allows me to simulate phonon modes in pi-stack arrays.  The method I use for generating these phonons is shown below:



When the phonon is mediated by the arenes, the energy levels take on a parabolic dispersion curve!  The origin of this curve is the nonlinearity of molecular orbital energies relative to inter-ring distances.  The phonon produces two different inter-ring distances in the array, X1 and X2.  X1, the shorter distance, produces much smaller HOMO-LUMO separations than X2, the longer distance, does.  The rate of change towards X1 is greater than that towards X2 during the phonon movement.  Thus, I have a finite effective mass for phonon-mediated transport through pi-stacks.  The dispersion curve example I attached is for fluorobenzene, a typical example.  I have observed this feature for many arenes and heterocycles.  I can change where the dispersion curve goes by adding ionic functional groups to the rings, inducing antiaromaticity in the molecules, making the molecules radicals, or turning the molecules into pairs of cation/anion heterocycles.  Elemental substitution, neutral functional groups, and mixing neutral species do not affect the direction of the typical dispersion curve, but they do change effective masses and initial molecular orbital energies.

Elasticity in lattices is essential for phonon-electron interactions, and since I have band gap narrowing under phonon movement, the array is not stable towards local dimer formation.  This means that the phonon must leave the array somehow, once it has entered the system.  It is not trapped efficiently by the arenes, because the energy variation is simply too small.  Calculations using statistical thermodynamics have shown that the average temperature required for a 0.2 angstrom disturbance is well below 100 kelvins for most arene or heterocycle arrays.  I have also verified the quality of fit for the parabolic curves using student t-test methods, showing that the dE2/dk2 term and hence effective mass obtained from it are very statistically significant.

What is your take on these findings?  I feel that they do explain previous findings done with DNA, which one of the molecules I tested is a good nucleobase mimic.

[voves]

Hi again,
I am calculating hole eff. masses in the strained InSb. If I understand, your script calculates long. and transv. eff. mass in the 100 and 010  (and 001) directions along A,B, and C vectors. But to calculate eff. mass in a warped band we need to average along 111 as well? Could you explain it or update the script, please?

1st and 2nd (N=1,2) eff. masses are almost the same for the unstrained InSb from your script (the same as for your Si example).

Jan

If you want to calculate the the effective mass of the electrons and holes in a semi-conductor,
I have created a small script for helping your in doing so. I have attached it to this post.

The script has been updated in a new improved version for support of anisotropic and/or indirect band gap semi-conductors.
And I have modified the tutorial to be on Si, since I could find experimental value of this material.
( However it should be noted, that one must manually enter k-points around which it should evaluate the effective mass,
and if the minimum is present at several places in the brillouin zone, the constant MC and MH must be adjusted represent this.
For instance in Si the lowest point of the conducting band is located 6 places in the brillouin zone, and hence MC must be set to 6)

Simple perform a calculation, and store the results in a NetCDF file,
and edit the script effective_mass.py to point at this netcdf file, and the run it as normal:
atk effective_mass.py

It will produced the following output ( based on a quick Si calculation i maded):

----------------------------------------------------------------------
# Electron effective mass
# ---> E                     = 0.238248813366
# ---> Conductivity          = 0.25471464737 me
# ---> Density of states     = 1.17006025908 me
# ---> Longitudinal          = 1.3419675337 me
# ---> Transverse (1)        = 0.165647613285 me
# ---> Transverse (2)        = 0.200168158579 me
----------------------------------------------------------------------
# Hole effective mass (1)
# ---> E                     = -0.23358258646
# ---> Conductivity          = 0.641102831335 me
# ---> Density of states     = 0.641102831335 me
# ---> Longitudinal          = 0.641102831305 me
# ---> Transverse (1)        = 0.64110283176 me
# ---> Transverse (2)        = 0.64110283094 me
----------------------------------------------------------------------
# Hole effective mass (2)
# ---> E                     = -0.233582586849
# ---> Conductivity          = 0.64110283086 me
# ---> Density of states     = 0.64110283086 me
# ---> Longitudinal          = 0.64110283089 me
# ---> Transverse (1)        = 0.641102830435 me
# ---> Transverse (2)        = 0.641102831255 me
----------------------------------------------------------------------
# Hole effective mass (3)
# ---> E                     = -0.233582587149
# ---> Conductivity          = 0.0926545669658 me
# ---> Density of states     = 0.0926545669658 me
# ---> Longitudinal          = 0.0926545669658 me
# ---> Transverse (1)        = 0.0926545669658 me
# ---> Transverse (2)        = 0.0926545669658 me
----------------------------------------------------------------------

Silicon has density of states effective mass of 1.08, conductivity effective mass of 0.26, and average hole mass of 0.56,
and I have collected the numbers in a small table below, and I think the agreement is pretty good.

Effective Mass	Experimental	Calculation	Error
Density of States	1.08	1.17	0.09
Conductivity	0.26	0.254	0.006
Hole (average)	0.56	0.46	0.1

[Nordland]

I will try to look into how to modify the script

[Nordland]

I think you can do the following, if the effective mass has to be evaluated around the Gamma point.

Instead of giving the exact Gamma point, you will give the value (1e-6,1e-6,1e-6), then it will still fit the effective mass to the Gamma point, but the longitudinal effective mass will be in the
direction of 111.

 

[voves]

Thanks, but I´m afraid, that the second derivatives will be calculated still along the A,B,C reciprocal lattice vectors.
I´m going to try it.

Jan

[Nordland]

The second derivative for the longitudinal effective mass is calculated along the line from the Gamma point to the given in the script,
and it will then make a polynomial fit around given point to get the effective mass.

Therefore if the value (1e-6,1e-6,1e-6) is given then the longitudinal effective mass will be evaluate in this direction. The transverse effective mass will
then be the directions perpendicular to this direction.

So I think it should work.

[Anders Blom]

Some caution is required. The 111 direction is not necessarily the same as (1,1,1) in units of the primitive reciprocal lattice vectors. The "111" direction refers to the corresponding conventional system, thus in each case one needs to consider which combination of the primitive vectors that make up this "Cartesian" direction. For fcc, it is towards the X-point at (1,1,1)/2, i.e. the relevant point to use in this script would rather be (1e-6,0,1e-6).

[voves]

Thanks, I found a script in the band structure tutorial. It shows the symetry points very well.

[Anders Blom]

Please note that the symmetry points for tIn (body-centered tetragonal) and the orthorhombic ones are under evaluation. I just found some mistakes in them. I will post an update as soon as possible.