Effective mass of electrons in Si  [SOLVED]

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Effective mass of electrons in Si

Postby Vladimir » Mon Aug 29, 2016 9:24 am

Dear all.

I'am trying to calculate Effective mass of electrons in Si.

my input file is Si.in and PP is silicon, fhi98PP : Trouiller-Martins-type, LDA Ceperley/Alder Perdew/Wang (1992)

I've scan eigenenergies around conduction band minima (0, 0.428936, 0.428936) and convert it to gnuplot readable format (scan.in) file.

using gnuplot I'am fitting this data with quadratic form with guplot script

Code: Select all
#!/usr/bin/gnuplot -persist

set grid

f(x,y,t) = Cxx*(x-a)**2 + Cyy*(y-b)**2 + Czz*(t-c)**2 + Cxy*(x-a)*(y-b) + Cxz*(x-a)*(t-c) + Cyz*(y-b)*(t-c) + E

Cxx=50.0; Cyy=50.0; Czz=50.0; Cxy=-10.0; Cxz=-10.0; Cyz=-100.0; a=0.0; b=0.428; c=0.428; E=6.72

fit f(x,y,t) "scan.in" using 1:2:3:8:(1) via Cxx, Cyy, Czz, Cxy, Cxz, Cyz, b, c, E

splot "scan.in" using 2:3:8 with linespoints, f(-0.05, x, y), f(0.0, x, y), f(0.05, x, y)


final set of parameters is:

Code: Select all
Final set of parameters            Asymptotic Standard Error
=======================            ==========================

Cxx             = 51.757           +/- 0.4894       (0.9456%)
Cyy             = 57.9317          +/- 2.208        (3.811%)
Czz             = 57.9317          +/- 2.208        (3.811%)
Cxy             = -9.06337         +/- 0.8243       (9.095%)
Cxz             = -9.06337         +/- 0.8243       (9.095%)
Cyz             = -101.408         +/- 1.913        (1.886%)
b               = 0.428936         +/- 0.0007661    (0.1786%)
c               = 0.428936         +/- 0.0007661    (0.1786%)
E               = 6.72609          +/- 0.001339     (0.01991%)


To diagonalize this quadratic form I am using numpy library and python script

Code: Select all
#!/usr/bin/env python

import numpy as np


# Cxx             = 51.757           +/- 0.4894       (0.9456%)
# Cyy             = 57.9317          +/- 2.208        (3.811%)
# Czz             = 57.9317          +/- 2.208        (3.811%)
# Cxy             = -9.06337         +/- 0.8243       (9.095%)
# Cxz             = -9.06337         +/- 0.8243       (9.095%)
# Cyz             = -101.408         +/- 1.913        (1.886%)

M = np.array(
    [
        [      51.757, -9.06337/2.0, -9.06337/2.0],
        [-9.06337/2.0,      57.9317, -101.408/2.0],
        [-9.06337/2.0, -101.408/2.0,      57.9317]
    ]
)

w, v = np.linalg.eig(M)

print w
print v


final eigenvalues of quadratic form is [ 52.6610137 6.3236863 108.6357 ]

How can I convert this values to:
Effective electron masses ml 0.98mo
Effective electron masses mt 0.19mo

Thanks in advance, Vladimir.
Attachments
Si.in
(1.56 KiB) Downloaded 254 times
scan.in
(8.88 KiB) Downloaded 261 times
Vladimir
 
Posts: 13
Joined: Sun Jun 05, 2016 5:22 am

Re: Effective mass of electrons in Si

Postby Vladimir » Mon Aug 29, 2016 10:17 am

maybe step in k-space is too large. I'am just decrease it and have got another values [ 60.34343569 9.61141188 103.61805243] and minima in (0, 0.42157, 0.42157)

even smaller step gave me [ 9.57368002 60.03041694 107.28150304] and minima in (0, 0.421598, 0.421598)

so lattice constant of Si is lattice_constant=5.43075e-10 m and rpim (0.0, 0.5, 0.5) length is lattice_constant/sqrt(2)

G = 2*pi / ( lattice_constant/sqrt(2) )

and free electrons E(ev) = (h*G)^2 / (m * e * 2) * k^2

where (h*G)^2 / (m * e * 2) = 10.2131559105 ev, and Effective electron masses are 1.06679520228 0.170133016398 0.0951995975196 ?

What does it means? Why there are three different masses?
Last edited by Vladimir on Tue Aug 30, 2016 9:21 am, edited 1 time in total.
Vladimir
 
Posts: 13
Joined: Sun Jun 05, 2016 5:22 am

Re: Effective mass of electrons in Si  [SOLVED]

Postby Vladimir » Tue Aug 30, 2016 9:01 am

yeh! SOLVED.

problem is that rprim is not orthogonal vectors

Code: Select all
rprim  0.0  0.5  0.5   # FCC primitive vectors (to be scaled by acell)
       0.5  0.0  0.5
       0.5  0.5  0.0


Orthogonal combinations are:
Code: Select all
 
G1 = -rprim[1] + rprim[2] + rprim[3]
G2 =  rprim[1] - rprim[2] + rprim[3]
G3 =  rprim[1] + rprim[2] - rprim[3]


So fit against this new variables with gnuplot

Code: Select all
#!/usr/bin/gnuplot -persist

set grid

FIT_LIMIT = 1e-11

f(x,y,t) = Cxx*(-x+y+t-a)**2 + Cyy*(x+t-y-b)**2 + Czz*(x+y-t-c)**2 + Cxy*(-x+y+t-a)*(x-y+t-b) + Cxz*(-x+y+t-a)*(x+y-t-c) + Cyz*(x-y+t-b)*(x+y-t-c) + E

Cxx=20.0; Cyy=20.0; Czz=20.0; Cxy=1.0; Cxz=1.0; Cyz=1.0; a=0.84; b=0.01; c=0.01; E=5.88

fit f(x,y,t) "scan.dat" using 1:2:3:8:(1) via Cxx, Cyy, Czz, Cxy, Cxz, Cyz, a, b, c, E

splot "scan.dat" using 2:3:8 with linespoints, f(-0.002, x, y), f(-0.001, x, y), f(0.0, x, y), f(0.001, x, y), f(0.002, x, y)


gave

Code: Select all
Cxx             = 5.33214          +/- 0.06533      (1.225%)
Cyy             = 27.3786          +/- 0.06533      (0.2386%)
Czz             = 27.3786          +/- 0.06533      (0.2386%)
Cxy             = -0.0678567       +/- 0.09819      (144.7%)
Cxz             = -0.0678568       +/- 0.09819      (144.7%)
Cyz             = -0.0249994       +/- 0.09819      (392.8%)
a               = 0.844206         +/- 2.984e-05    (0.003535%)
b               = 3.83067e-06      +/- 4.673e-06    (122%)
c               = 3.83067e-06      +/- 4.673e-06    (122%)
E               = 5.88797          +/- 5.915e-07    (1.005e-05%)


Cxy, Cxz, Cyz, b and c vanish.

G = 2*pi / lattice_constant

and masses [ 0.9498737 0.18629819 0.18636041] with good agreements with experiment.
Vladimir
 
Posts: 13
Joined: Sun Jun 05, 2016 5:22 am

Re: Effective mass of electrons in Si

Postby mverstra » Sat May 20, 2017 12:00 am

Excellent. Also note that in the latest version of abinit you can calculate effective masses analytically through the second k derivative. See the release notes, and more stuff is coming in version 8.4 in a month or so (~ June 2017?)
Matthieu Verstraete
University of Liege, Belgium
mverstra
 
Posts: 637
Joined: Wed Aug 19, 2009 12:01 pm


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