Fitting of $F_{res} vs. H$ for pMTJ structures using Python

##Introduction

In this project, we ae going to model an all prependicular device based on Kittle formula and try to calculate its FMR frequency under the application of a tilted field. The structure contains two magnetic layers, $1$ is the fixed layer and $2$ is the free layer, which are separated by an insulator spacer. Based on Kittle formula, the resonance frequency, $\omega_r$ , for an applied DC magnetic field, $H_{dc}$ , is equal to

$\displaystyle(\frac{\omega_r}{\gamma})^2=H_1 H_2$

$\left\{ \begin{array}{ll} \displaystyle H_1=H_{dc}\ cos(θ_H−θ_2)+H_{st}\ cos(θ_2−θ_1)−H_{K21}^{eff}\ cos^2θ_2−H_{K22}\ cos^4θ_2 \\ \displaystyle H_2=H_{dc}\ cos(θ_H-θ_2)+H_{st}\ cos(θ_2−θ_1)−(H_{K21}^{eff}+\frac{H_{K22}}{2})\ cos2θ_2−\frac{H_{K22}}{2}\ cos 4θ_2 \end{array} \right.$

Here, $\gamma$ is Gyromagnetic ratio, $\displaystyle H_{Ki1}^{eff} = \frac{M_s}{\mu_0} - 2*\frac{K_{i1}+2K_{i2}}{M_s}$ and $\displaystyle H_{Ki2}^{eff} = 4*\frac{K_{i2}}{M_s}$ are the magnetic anisotropy fields in which $K_{i1}$ and $K_{i2}$ are the first and second order anisotropy for the $i_{th}$ layer. $H_{st}$ is defined as the stray field between the layers. $θ_H$ is the angle of the DC applied field and $θ_1$ and $θ_2$ are the equilibrium angles of the layers when the DC field is applied to the structure. All the angles are defined w.r.t. the normal vector of the layers. To find $θ_1$ and $θ_2$ values, the equilibrium conditions of both the layers should be solved for all the applied field values.

$H_{dc}\ sin(θ_H − θ_{eq1})−H_{st}\ sin(θ_{eq1}−θ_{eq2})+H_{K11}^{eff}\ sinθ_{eq1}\ cosθ_{eq1}+H_{K12}\ sinθ_{eq1}\ cos^3θ_{eq1}=0$

$H_{dc}\ sin(θ_H − θ_{eq2})−H_{st}\ sin(θ_{eq2}−θ_{eq1})+H_{K21}^{eff}\ sinθ_{eq2}\ cosθ_{eq2}+H_{K22}\ sinθ_{eq2}\ cos^3θ_{eq2}=0$

In order to simplify the model, we consider the fixed layer is always prependicular ( $θ_{eq1} = 0$ ), and also ignore the second order anisotropy of the free layer ( $H_{K22} = 0$ ). Then the equilibrium condition for the free layer is:

$H_{dc}\ sin(θ_H − θ_{eq2})−H_{st}\ sin(θ_{eq2})+H_{K21}^{eff}\ sinθ_{eq2}\ cosθ_{eq2}=0$

Also $H_{1}$ and $H_{2}$ is simpliefied to

$\left\{ \begin{array}{ll} \displaystyle H_1=H_{dc}\ cos(θ_H−θ_2)+H_{st}\ cos(θ_2)−H_{K21}^{eff}\ cos^2θ_2 \\ \displaystyle H_2=H_{dc}\ cos(θ_H-θ_2)+H_{st}\ cos(θ_2)−(H_{K21}^{eff})\ cos2θ_2 \end{array} \right.$

Using the formula, we can simply extract the equilibrium angle of the magnetization versus the strength of the field for a given external field angel.

##Results

First we import the $Python$ libraries we need. $Numpy$ for handling arrays, $Scipy$ for finding the equilibrium angle, and $pylab$ for plotting.

In [1]:

from scipy.optimize import fsolve
import math
import numpy as np
from scipy.optimize import curve_fit
from numpy import *
import matplotlib.pyplot as plt
import matplotlib.lines as mlines

import pylab as pl           # Imports all plotting capability of magplotlib.
%matplotlib inline           
# "%matplotlib inline" makes all plots show up inside the document (i.e. "inline") and not in a separate plot window.

Then we need to set the ranges for the parameters in our model. Here we can define the ranges for $H_{K11}$ and $Hst$ . The code uses for loops to solve the equation for every combinations of these two parameters.

In [2]:

Hk21max=16000
Hk21min=16000
Hk21step=500 # should be positive
#----------------------------------
Hstmax=0
Hstmin=0
Hststep=500 # should be positive
#==================================
gamma = 0.00319

Here, the external field is defined. $angle$ is the angle of the applied DC magnetic field in degree w.r.t. the normal vector of the layers (For in-plane field, $angle = 90$ ), and $H_{dc}$ is vector contains the strength of the applied field.

In [3]:

angle=20

#Whole the range
Hdc = array(range(0, 20000, 50))

Here the equilibrium condition and the resonance frequency for the free layer based on Kittel formula are defined.

In [4]:

def equilibrium_eq(p):
    theta2 = p
    return (H*math.cos(theta2+np.pi/2-thetaH)-Hst*math.sin(theta2)+Hk21*math.sin(theta2)*math.cos(theta2))
################################################################################
def fres_func(Hdc, theta2):
    return gamma*( (Hdc*np.sin(theta2+np.pi/2-thetaH)+Hst*np.cos(theta2)-(HK21)*np.cos(2*theta2)) * (Hdc*np.sin(theta2+np.pi/2-thetaH)+Hst*np.cos(theta2)-HK21*np.cos(theta2)**2) )**(0.5)

The next cell is for the initialization of the variables used in the code.

In [5]:

thetaH=np.pi/180*(angle)
Theta2 = array(zeros(len(Hdc)))
Frequency = array(zeros(len(Hdc)))

Here, based on the ranges defined for the parameters, the code does the sweep and fitting. The output is the table contains the combination of the parameters and the calculated $adj.r.sq$ values.

In [6]:

print "=========================================================================================================================="
print "----------------------------------------------------------------------------"
print "gamma=",gamma,", angle=",angle
print "Hdc :", (Hdc)
print "----------------------------------------------------------------------------"
plt.close('all')
print "=========================================================================================================================="
for Hst in xrange(Hstmin,Hstmax+Hststep,Hststep):
    for Hk21 in xrange(Hk21min,Hk21max+Hk21step,Hk21step):
        HK21=[Hk21]
        for i in xrange(0,len(Hdc)):
            H = Hdc[i]
            x2 =  fsolve(equilibrium_eq, (0))
            Theta2[i]=x2
################################################################################
        Frequency = fres_func(Hdc, Theta2)
        print "Hst=",np.round(Hst),", Hk21=",np.round(Hk21)
        print "--------------------------------------------------------------------------------------------------------------------------"
        print "theta :", (Theta2/np.pi*180)
        print "=========================================================================================================================="
        print "--------------------------------------------------------------------------------------------------------------------------"
        print "Frequency :", (Frequency)
        print "=========================================================================================================================="
        plt.figure("Theta", figsize=(8, 6), dpi=80)
        plt.ylabel('theta (degree)')
        plt.xlabel('Hdc (Oe)')
        plt.plot(Hdc, Theta2/np.pi*180)
        #plt.plot(Hdc,  angel, 'g^',label='Field angle')
        #plt.title('Theta1:blue, Theta2:red, Relative Theta:green ')
        #plt.legend(loc='upper left',handles=[mlines.Line2D([], [], color='blue', marker='s', label='Theta1'),mlines.Line2D([], [], color='red', marker='s', label='Theta2'),mlines.Line2D([], [], color='green', marker='^', label='relTheta')],numpoints=1)       
        plt.figure("main", figsize=(8, 6), dpi=80)
        ax = plt.subplot(111)
        plt.ylabel('f_resonance (GHz)')
        plt.xlabel('Hdc (Oe)')
        plt.plot(Hdc, Frequency)
        #plt.plot(Hdc, fittedFres, label=('Hk21='+str(Hk21)+' ,Hst='+str(Hst)) )

==========================================================================================================================
----------------------------------------------------------------------------
gamma= 0.00319 , angle= 20
Hdc : [    0    50   100   150   200   250   300   350   400   450   500   550
 650   700   750   800   850   900   950  1000  1050  1100  1150
1250  1300  1350  1400  1450  1500  1550  1600  1650  1700  1750
1850  1900  1950  2000  2050  2100  2150  2200  2250  2300  2350
2450  2500  2550  2600  2650  2700  2750  2800  2850  2900  2950
3050  3100  3150  3200  3250  3300  3350  3400  3450  3500  3550
3650  3700  3750  3800  3850  3900  3950  4000  4050  4100  4150
4250  4300  4350  4400  4450  4500  4550  4600  4650  4700  4750
4850  4900  4950  5000  5050  5100  5150  5200  5250  5300  5350
5450  5500  5550  5600  5650  5700  5750  5800  5850  5900  5950
6050  6100  6150  6200  6250  6300  6350  6400  6450  6500  6550
6650  6700  6750  6800  6850  6900  6950  7000  7050  7100  7150
7250  7300  7350  7400  7450  7500  7550  7600  7650  7700  7750
7850  7900  7950  8000  8050  8100  8150  8200  8250  8300  8350
8450  8500  8550  8600  8650  8700  8750  8800  8850  8900  8950
9050  9100  9150  9200  9250  9300  9350  9400  9450  9500  9550
9650  9700  9750  9800  9850  9900  9950 10000 10050 10100 10150
10250 10300 10350 10400 10450 10500 10550 10600 10650 10700 10750
10850 10900 10950 11000 11050 11100 11150 11200 11250 11300 11350
11450 11500 11550 11600 11650 11700 11750 11800 11850 11900 11950
12050 12100 12150 12200 12250 12300 12350 12400 12450 12500 12550
12650 12700 12750 12800 12850 12900 12950 13000 13050 13100 13150
13250 13300 13350 13400 13450 13500 13550 13600 13650 13700 13750
13850 13900 13950 14000 14050 14100 14150 14200 14250 14300 14350
14450 14500 14550 14600 14650 14700 14750 14800 14850 14900 14950
15050 15100 15150 15200 15250 15300 15350 15400 15450 15500 15550
15650 15700 15750 15800 15850 15900 15950 16000 16050 16100 16150
16250 16300 16350 16400 16450 16500 16550 16600 16650 16700 16750
16850 16900 16950 17000 17050 17100 17150 17200 17250 17300 17350
17450 17500 17550 17600 17650 17700 17750 17800 17850 17900 17950
18050 18100 18150 18200 18250 18300 18350 18400 18450 18500 18550
18650 18700 18750 18800 18850 18900 18950 19000 19050 19100 19150
19250 19300 19350 19400 19450 19500 19550 19600 19650 19700 19750
19850 19900 19950]
----------------------------------------------------------------------------
==========================================================================================================================
Hst= 0 , Hk21= 16000
--------------------------------------------------------------------------------------------------------------------------
theta : 

/ext/sage/sage-8.0/local/lib/python2.7/site-packages/scipy/optimize/minpack.py:161: RuntimeWarning: The iteration is not making good progress, as measured by the 
  improvement from the last ten iterations.
  warnings.warn(msg, RuntimeWarning)
/ext/sage/sage-8.0/local/lib/python2.7/site-packages/scipy/optimize/minpack.py:161: RuntimeWarning: The iteration is not making good progress, as measured by the 
  improvement from the last five Jacobian evaluations.
  warnings.warn(msg, RuntimeWarning)
/ext/sage/sage-8.0/local/lib/python2.7/site-packages/ipykernel/__main__.py:6: RuntimeWarning: invalid value encountered in sqrt

[  0.          -0.06141884  -0.1232006   -0.18534859  -0.24786615
  -0.31075667  -0.37402359  -0.43767038  -0.50170059  -0.56611778
  -0.63092559  -0.69612768  -0.7617278   -0.82772971  -0.89413724
  -0.96095429  -1.02818479  -1.09583274  -1.16390219  -1.23239726
  -1.30132212  -1.37068099  -1.44047818  -1.51071804  -1.58140499
  -1.65254352  -1.72413818  -1.79619361  -1.86871449  -1.94170559
  -2.01517175  -2.08911789  -2.16354899  -2.23847013  -2.31388645
  -2.38980319  -2.46622566  -2.54315926  -2.62060949  -2.69858191
  -2.7770822   -2.85611613  -2.93568956  -3.01580843  -3.09647882
  -3.17770689  -3.2594989   -3.34186123  -3.42480038  -3.50832294
  -3.59243563  -3.6771453   -3.7624589   -3.84838353  -3.9349264
  -4.02209486  -4.10989641  -4.19833866  -4.28742939  -4.37717652
  -4.46758812  -4.55867241  -4.65043778  -4.74289278  -4.83604614
  -4.92990675  -5.02448368  -5.11978619  -5.21582374  -5.31260597
  -5.41014272  -5.50844405  -5.60752024  -5.70738176  -5.80803934
  -5.90950393  -6.01178672  -6.11489916  -6.21885294  -6.32366005
  -6.42933271  -6.53588346  -6.64332512  -6.75167082  -6.860934
  -6.97112842  -7.08226818  -7.19436774  -7.30744191  -7.42150588
  -7.53657521  -7.6526659   -7.76979433  -7.88797732  -8.00723217
  -8.12757661  -8.24902888  -8.37160771  -8.49533236  -8.62022263
  -8.74629892  -8.87358218  -9.00209399  -9.13185661  -9.26289291
  -9.39522651  -9.52888175  -9.66388372  -9.80025833  -9.93803231
 -10.07723328 -10.21788976 -10.36003126 -10.50368827 -10.64889236
 -10.79567619 -10.94407361 -11.09411968 -11.24585076 -11.39930459
 -11.55452031 -11.7115386  -11.87040174 -12.03115369 -12.1938402
 -12.35850892 -12.52520949 -12.69399369 -12.86491557 -13.03803157
 -13.21340067 -13.39108461 -13.57114801 -13.7536586  -13.93868743
 -14.1263091  -14.31660202 -14.5096487  -14.70553603 -14.90435566
 -15.10620433 -15.31118432 -15.51940388 -15.73097775 -15.94602772
 -16.16468323 -16.38708214 -16.61337141 -16.84370807 -17.07826019
 -17.31720796 -17.56074502 -17.8090799  -18.06243766 -18.32106181
 -18.58521652 -18.85518916 -19.13129328 -19.41387206 -19.70330245 -20.
 -20.30442466 -20.61708773 -20.93856032 -21.26948363 -21.61058165
 -21.962677   -22.3267109  -22.7037686  -23.09511235 -23.50222474
 -23.92686671 -24.37115691 -24.83768258 -25.32965923 -25.85116775
 -26.40752092 -27.00585744 -27.65616464 -28.3731822  -29.18034196
 -30.11927711 -31.27928703 -32.95187039 -35.49412469 -35.46890554
 -35.3507399  -35.28226834 -35.20535311 -35.13004491 -35.05925216
 -34.98053943 -34.90904108 -34.83894403 -34.7513119  -34.68564584
 -34.61008223 -34.51183406 -34.45802045 -34.41675556 -34.3016975
 -34.23096385 -34.15025756 -34.07691338 -33.99492812 -33.92240638
 -33.84880687 -33.75983631 -33.66952318 -33.59594363 -33.51393875
 -33.41963573 -33.3683364  -33.22659285 -33.2181317  -33.08811115
 -33.01720251 -32.9451459  -32.85636357 -32.76611991 -32.69018686
 -32.59704349 -32.51036539 -32.42221273 -32.30802189 -32.19185398
 -32.17347224 -32.08763776 -31.99181196 -31.87718471 -31.82081564
 -31.71558087 -31.63473958 -31.53910623 -31.47744739 -31.34240217
 -31.20477365 -31.21171165 -31.07012389 -30.98204412 -30.89221215
 -30.78143291 -30.70707868 -30.58958968 -30.4958799  -30.3550117
 -30.29315593 -30.14849555 -30.14484339 -29.99540207 -29.92684105
 -29.79290707 -29.61290601 -29.58115995 -29.47161581 -29.35951079
 -29.25602276 -29.15005661 -29.04152365 -28.97706526 -28.8163711
 -28.72350531 -28.62827416 -28.50601245 -28.38055589 -28.28961481
 -28.1195439  -28.08735768 -27.94921129 -27.80720266 -27.6949336
 -27.59313615 -27.57872232 -27.30996613 -27.26222587 -26.98043926
 -26.985162   -26.84758545 -26.75131287 -26.62868808 -26.45479945
 -26.37108123 -26.23567637 -26.0955648  -25.95049676 -25.80878171
 -25.67933857 -25.62513894 -25.24246421 -25.28859859 -25.11033946
 -25.00181196 -24.8102989  -24.69089431 -24.40258169 -24.43635149
 -24.25772416 -24.09277294 -23.94246459 -23.76198186 -23.59609661
 -23.47001376 -23.31266481 -23.14693218 -22.94605241 -22.73386722
 -22.6473198  -22.49879678 -22.25330136 -22.11307363 -21.94806256
 -21.72215913 -21.90050284 -21.58870572 -21.25340838 -21.04107871
 -20.96675245 -20.60276948 -20.46012572 -20.17152685 -19.94760211
 -19.70075542 -19.53006411 -19.39423184 -19.1267504  -18.89447774
 -18.62165725 -18.38492943 -17.780573   -17.64119018 -17.59153705
 -17.35727781 -17.07452963 -16.81189035 -16.45596972 -16.32753396
  48.23897561  48.15748113  48.0762525   47.9952895   47.9145919
  47.83415947  47.75399198  47.67408918  47.59445081  47.51507662
  47.43596633  47.35711969  47.2785364   47.20021618  47.12215874
  47.04436377  46.96683097  46.88956003  46.81255062  46.73580242
  46.65931509  46.5830883   46.5071217   46.43141493  46.35596763
  46.28077945  46.20585     46.13117891  46.0567658   45.98261027
  45.90871192  45.83507036  45.76168518  45.68855595  45.61568226
  45.54306369  45.47069979  45.39859013  45.32673427  45.25513176
  45.18378214  45.11268494  45.04183971  44.97124597  44.90090325
  44.83081105  44.7609689   44.6913763   44.62203274  44.55293774
  44.48409077  44.41549133  44.3471389   44.27903295  44.21117295
  44.14355838  44.07618869  44.00906334  43.94218179  43.87554349
  43.80914787  43.74299438  43.67708246  43.61141154]
==========================================================================================================================
--------------------------------------------------------------------------------------------------------------------------
Frequency : [  5.10400000e+01   5.08900896e+01   5.07401194e+01   5.05900877e+01
   5.04399929e+01   5.02898334e+01   5.01396075e+01   4.99893135e+01
   4.98389496e+01   4.96885140e+01   4.95380049e+01   4.93874205e+01
   4.92367588e+01   4.90860180e+01   4.89351959e+01   4.87842907e+01
   4.86333002e+01   4.84822224e+01   4.83310551e+01   4.81797961e+01
   4.80284432e+01   4.78769940e+01   4.77254464e+01   4.75737978e+01
   4.74220459e+01   4.72701881e+01   4.71182221e+01   4.69661450e+01
   4.68139545e+01   4.66616476e+01   4.65092217e+01   4.63566739e+01
   4.62040015e+01   4.60512014e+01   4.58982706e+01   4.57452061e+01
   4.55920048e+01   4.54386634e+01   4.52851786e+01   4.51315472e+01
   4.49777657e+01   4.48238307e+01   4.46697384e+01   4.45154854e+01
   4.43610678e+01   4.42064818e+01   4.40517236e+01   4.38967891e+01
   4.37416742e+01   4.35863747e+01   4.34308865e+01   4.32752050e+01
   4.31193258e+01   4.29632444e+01   4.28069559e+01   4.26504557e+01
   4.24937388e+01   4.23368001e+01   4.21796345e+01   4.20222367e+01
   4.18646012e+01   4.17067226e+01   4.15485951e+01   4.13902129e+01
   4.12315700e+01   4.10726604e+01   4.09134776e+01   4.07540153e+01
   4.05942669e+01   4.04342256e+01   4.02738845e+01   4.01132363e+01
   3.99522739e+01   3.97909896e+01   3.96293758e+01   3.94674245e+01
   3.93051276e+01   3.91424768e+01   3.89794634e+01   3.88160787e+01
   3.86523136e+01   3.84881587e+01   3.83236045e+01   3.81586410e+01
   3.79932583e+01   3.78274458e+01   3.76611928e+01   3.74944882e+01
   3.73273208e+01   3.71596786e+01   3.69915497e+01   3.68229216e+01
   3.66537815e+01   3.64841162e+01   3.63139120e+01   3.61431549e+01
   3.59718303e+01   3.57999233e+01   3.56274183e+01   3.54542995e+01
   3.52805502e+01   3.51061534e+01   3.49310915e+01   3.47553462e+01
   3.45788986e+01   3.44017292e+01   3.42238177e+01   3.40451432e+01
   3.38656839e+01   3.36854173e+01   3.35043200e+01   3.33223678e+01
   3.31395355e+01   3.29557970e+01   3.27711251e+01   3.25854915e+01
   3.23988668e+01   3.22112205e+01   3.20225208e+01   3.18327344e+01
   3.16418267e+01   3.14497617e+01   3.12565016e+01   3.10620072e+01
   3.08662373e+01   3.06691489e+01   3.04706969e+01   3.02708342e+01
   3.00695113e+01   2.98666764e+01   2.96622749e+01   2.94562497e+01
   2.92485404e+01   2.90390837e+01   2.88278128e+01   2.86146571e+01
   2.83995422e+01   2.81823893e+01   2.79631152e+01   2.77416315e+01
   2.75178445e+01   2.72916548e+01   2.70629562e+01   2.68316361e+01
   2.65975739e+01   2.63606410e+01   2.61206995e+01   2.58776017e+01
   2.56311887e+01   2.53812897e+01   2.51277202e+01   2.48702808e+01
   2.46087554e+01   2.43429093e+01   2.40724869e+01   2.37972087e+01
   2.35167689e+01   2.32308310e+01   2.29390243e+01   2.26409380e+01
   2.23361158e+01   2.20240483e+01   2.17041641e+01   2.13758191e+01
   2.10382831e+01   2.06907232e+01   2.03321828e+01   1.99615551e+01
   1.95775484e+01   1.91786412e+01   1.87630217e+01   1.83285065e+01
   1.78724269e+01   1.73914676e+01   1.68814305e+01   1.63368763e+01
   1.57505590e+01   1.51124857e+01   1.44082522e+01   1.36158423e+01
   1.26987404e+01   1.15884777e+01   1.01268415e+01   7.72621119e+00
   1.62010258e-01              nan   1.30794228e-01              nan
   1.51152630e-01   2.64972003e-01   1.24417928e-01   3.44954625e-01
   2.55192043e-01              nan   4.88387401e-01   2.13550225e-01
   2.17123393e-01   7.03383174e-01   2.17404476e-01              nan
   3.19578189e-01              nan   1.80136550e-01              nan
   9.88037868e-02              nan              nan              nan
   3.90260013e-01   1.60232953e-01   2.16669861e-01   5.31820416e-01
              nan   8.94974897e-01              nan   5.61934988e-01
   3.10673440e-01              nan              nan   2.56742887e-01
              nan   1.91537821e-01   2.15408768e-01   2.69228137e-01
   7.17846355e-01   9.87548998e-01              nan              nan
              nan   5.59223606e-01              nan   1.30897451e-01
              nan              nan              nan   3.21872964e-01
   8.69130063e-01              nan              nan              nan
              nan   7.41047402e-02              nan              nan
              nan   7.02236866e-01              nan   7.64684894e-01
              nan              nan              nan              nan
   9.45194925e-01              nan   2.36630953e-01   3.73253244e-01
   3.30292655e-01   3.16348477e-01   3.35340545e-01              nan
   4.49743569e-01   1.00967372e-01              nan              nan
   3.27941399e-01              nan   6.83266944e-01              nan
              nan   3.13028329e-01   2.11971296e-01              nan
              nan   5.50736748e-01              nan   1.04860943e+00
              nan   1.53420369e-01              nan              nan
   3.69395343e-01              nan              nan              nan
              nan   2.14021254e-01   4.31012944e-02              nan
   1.11423454e+00              nan   3.44151812e-01              nan
   4.69067574e-01   1.99375186e-01   1.00200629e+00              nan
              nan   2.35885585e-01   1.93587015e-01   4.40796565e-01
   4.95522425e-01   2.22266431e-01   1.73541105e-01   2.17660917e-01
   4.94871730e-01   6.93902908e-01   1.99689334e-01              nan
   5.29664444e-01   3.15826831e-01   1.80183544e-01   4.84278125e-01
              nan              nan              nan              nan
              nan   2.72189400e-01              nan   3.75423649e-01
   4.22897945e-01   5.24380306e-01   3.30449601e-01              nan
              nan              nan   1.92241608e-01   1.05221011e-01
   9.21501707e-01   7.33062959e-01   2.03394141e-01              nan
              nan              nan   2.96458266e-02              nan
   3.60786269e+01   3.61662989e+01   3.62541536e+01   3.63421918e+01
   3.64304141e+01   3.65188212e+01   3.66074137e+01   3.66961921e+01
   3.67851571e+01   3.68743092e+01   3.69636490e+01   3.70531771e+01
   3.71428940e+01   3.72328002e+01   3.73228961e+01   3.74131824e+01
   3.75036595e+01   3.75943279e+01   3.76851879e+01   3.77762402e+01
   3.78674850e+01   3.79589228e+01   3.80505540e+01   3.81423791e+01
   3.82343983e+01   3.83266120e+01   3.84190207e+01   3.85116246e+01
   3.86044240e+01   3.86974193e+01   3.87906108e+01   3.88839987e+01
   3.89775834e+01   3.90713651e+01   3.91653441e+01   3.92595206e+01
   3.93538948e+01   3.94484669e+01   3.95432372e+01   3.96382058e+01
   3.97333730e+01   3.98287388e+01   3.99243035e+01   4.00200672e+01
   4.01160301e+01   4.02121922e+01   4.03085536e+01   4.04051146e+01
   4.05018751e+01   4.05988352e+01   4.06959950e+01   4.07933546e+01
   4.08909140e+01   4.09886733e+01   4.10866325e+01   4.11847915e+01
   4.12831504e+01   4.13817092e+01   4.14804679e+01   4.15794264e+01
   4.16785848e+01   4.17779429e+01   4.18775007e+01   4.19772582e+01]
==========================================================================================================================

/ext/sage/sage-8.0/local/lib/python2.7/site-packages/matplotlib/font_manager.py:273: UserWarning: Matplotlib is building the font cache using fc-list. This may take a moment.
  warnings.warn('Matplotlib is building the font cache using fc-list. This may take a moment.')

Fitting of Fresvs.HF_{res} vs. HFres​vs.H for pMTJ structures using Python

Fitting of $F_{res} vs. H$ for pMTJ structures using Python