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## Effective mass of electrons in Si [SOLVED]

Moderators: jzwanzig, jolafc

Posts: 13
Joined: Sun Jun 05, 2016 5:22 am

### Effective mass of electrons in Si

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

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``#!/usr/bin/gnuplot -persistset gridf(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) + ECxx=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=6.72fit f(x,y,t) "scan.in" using 1:2:3:8:(1) via Cxx, Cyy, Czz, Cxy, Cxz, Cyz, b, c, Esplot "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:

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``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

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``#!/usr/bin/env pythonimport 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 wprint 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

Attachments Si.in scan.in

Posts: 13
Joined: Sun Jun 05, 2016 5:22 am

### Re: Effective mass of electrons in Si

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.

Posts: 13
Joined: Sun Jun 05, 2016 5:22 am

### Re: Effective mass of electrons in Si  [SOLVED]

yeh! SOLVED.

problem is that rprim is not orthogonal vectors

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``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:

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`` G1 = -rprim + rprim + rprimG2 =  rprim - rprim + rprimG3 =  rprim + rprim - rprim``

So fit against this new variables with gnuplot

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``#!/usr/bin/gnuplot -persistset gridFIT_LIMIT = 1e-11f(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) + ECxx=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=5.88fit f(x,y,t) "scan.dat" using 1:2:3:8:(1) via Cxx, Cyy, Czz, Cxy, Cxz, Cyz, a, b, c, Esplot "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

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``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.

mverstra
Posts: 655
Joined: Wed Aug 19, 2009 12:01 pm

### Re: Effective mass of electrons in Si

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