software to fit optical spectra - Quantum Materials Group

More documents

Recommendations

Info

ε Equation 2-22 N i, ∧ var ( ω) = ∑ Aiε ( ω) i= 1 and consider coefficients A i as free parameters. To ensure that ε 2 ( ω) ≥ 0 , we require that all A ≥ 0 . It is convenient to set them to zero at the boundaries: A A = 0 , ensuring that i ε 2 ( ω) vanishes at 1 ω and N ω 8 . 1 = N As is schematically shown on Figure 2-4, the imaginary part of ε var ( ω) is a piecewise curve going through N points { ω i , Ai }(i = 1,..,N). One can see that Imε var ( ω) in between the reference frequency points ω i is simply given by a linear interpolation. The Reε var ( ω) is its exact KK transform. The free parameters of ε var ( ω) are the values of ε 2 at every frequency point ω i . There are altogether N − 2 parameters. Obviously, this construction is extremely flexible and totally model-independent, just what we aimed to obtain! Dielectric function ε 0 ω 1 ω i-1 ω i ω i+1 Frequency ω N Im ε var Re ε var ε i-1,Λ Figure 2-4. A KK-constrained variational dielectric function. The Imε var ( ω) is composed of many triangular functions Im ( ) , ε ω ∧ i and the Reε var ( ω) is the exact KK-transform of Imε var ( ω) . The second problem is that the so-constructed variational dielectric function (Equation 2-22) totally ignores all spectral weight beyond the frequency range [ ω min , ωmax ] . However, according to the KK relation (Equation 2-17), the non-zero ε 2 ( ω) outside this region influences ε 1( ω) inside it, and is, therefore, essential to calculate reflectivity R (ω) , which depends on both ε ( ) and ε ( ) . 1 ω 2 ω 8 a non-zero value of ε 2 at the boundary would result in a discontinuity of ε 2 and therefore in an unwanted divergence of ε 1 ε i,Λ ε i+1,Λ Guide to RefFIT Page 20
The problem can be circumvented by doing the fitting in two steps. Initially, the spectra ( 0) are fitted in a conventional way by some formula-defined dielectric function ε ( ω) = ε mod ( ω) . If the match is reasonably good, then we can assume that ε mod ( ω) has correct frequency dependence outside the considered spectral range, even though some fine details of the experimental curve are not fitted very well. Then we can fix all parameters of ε mod ( ω) and add a variational function to it: ( 1) ε ω) = ε ( ω) + ε ( ω) . ( mod var Now the ε var ( ω) acts as a small correction to the initial model ε mod ( ω) . When we do a ( 1) variational fitting of the reflectivity spectrum with ε ( ω) , the ‘KK influence’ of the low- and high-frequency spectral weights on ε 1( ω) inside [ ω min , ωmax ] is already accounted for by ε ( ) . mod ω Because ε var ( ω) is now added to ε mod ( ω) , which is the dominant contribution to the total dielectric function, the parameters A i are not necessarily positive (which was initially required by the condition ε ( ω) ≥ 0 ), but can be negative as well. 2 From the Equation 2-22 we can get a simple analytical formula for the first derivatives of the dielectric function ε var ( ω) with respect to the parameters A i , which are required by the Levenberg-Marquardt procedure (section 2.1.1): ∂ε i ∧ var ( ω) , = ε ∂A i ( ω) . At the end we minimize the chi-square with respect to all N − 2 parameters, thus obtaining ‘the best’ KK-related dielectric function, which fits the reflectivity spectrum. In the above example we discussed the fitting of a single reflectivity spectrum. As was mentioned before, in the particular case of a normal-incidence reflectivity spectrum of an isotropic sample, (almost) the same result could be obtained by the ‘conventional’ KK technique. However, an important advantage of the new method is that it can be applied to virtually any type of optical spectra, or a combination of them! The typical number of parameters that are adjusted in the fitting by variational functions is very large – up to the number of experimental points, which might amount to few thousands. Although such an enormous number of parameters seem to make the fitting procedure prohibitively slow, we found that it nevertheless converges within acceptable time limits. This is yet another reason to admire, how good the LM method is! In conclusion, it is worth mentioned that the described method is being successfully utilized in our group. However, there is a set of not yet well-addressed issues, for instance, a sensitivity of the output dielectric function to the noise and systematic error bars in the input data, or the optimal number and distribution of the frequencies ω i forming the mesh. Until we know more about the possible caveats, one should be especially careful when doing this kind of analysis and interpreting the output data. Guide to RefFIT Page 21
Page 1 and 2: Last updated: 20.04.2004 Guide to R
Page 3 and 4: 4.8. Dataset manager ..............
Page 5 and 6: that time. The later development of
Page 7 and 8: 1.4. Let’s keep in touch Please,
Page 9 and 10: 2.1.1. Levenberg-Marquardt algorith
Page 11 and 12: Guess initial parameters p , , p 1
Page 13 and 14: achieve a proper ‘balance’ betw
Page 15 and 16: The third requirement is that at ve
Page 17 and 18: Equation 2-19 is useful to estimate
Page 19: The first problem is that the Loren
Page 23 and 24: In these cases the fitting of the d
Page 25 and 26: 3. Tutorial This chapter shows how
Page 27 and 28: Figure 3-3 Graph contents window is
Page 29 and 30: Figure 3-7 Click the button and in
Page 31 and 32: If you are still far from this fitt
Page 33 and 34: Figure 3-15 The model and experimen
Page 35 and 36: Figure 3-18 The “Graph properties
Page 37 and 38: Figure 3-20 Loading the reflectivit
Page 39 and 40: Figure 3-22 The general RefFIT view
Page 41 and 42: Figure 3-25 The general RefFIT view
Page 43 and 44: MainWindow(XPos = 0, YPos = 0, Widt
Page 45 and 46: emarkable ‘anomaly’, related to
Page 47 and 48: ###################################
Page 49 and 50: # restores the original application
Page 51 and 52: 4. Reference 4.1. System requiremen
Page 53 and 54: Figure 4-2 The “Parameter control
Page 55 and 56: The model file stores information a
Page 57 and 58: considering multiple (Fabri- Perot)
Page 59 and 60: optical conductivity in practical u
Page 61 and 62: Kronig friendly, so be careful! “
Page 63 and 64: function. This command cannot be us
Page 65 and 66: Iinc Itrans ε ε ( 1) ( 2) … (n)
Page 67 and 68: In ellipsometry the complex ratio
Page 69 and 70: ` Figure 4-10 The experimental conf
Page 71 and 72:
Here ω p and ∞ ε are considered
Page 73 and 74:
points “#pts” and the grid type
Page 75 and 76:
4.9.2. Graph contents To set the gr
Page 77 and 78:
Available ChiSq terms and click the
Page 79 and 80:
At each iteration a linear equation
Page 81 and 82:
Before a macro is actually executed
Page 83 and 84:
Quantity Text Abbreviation of the q
Page 85 and 86:
Parameters: Name Type Description W
Page 87 and 88:
“Wp” - plasma frequency; “G
Page 89 and 90:
ShowLine Integer =12: Chalk =13: Br
Page 91 and 92:
XMin Double Minimal X (frequency) v
Page 93 and 94:
always come in pairs. All commands
show all

software to fit optical spectra - Quantum Materials Group

Create successful ePaper yourself

Delete template?

Save as template?