JP5949567B2 - Method and apparatus for calculating optical band gap of semiconductor material - Google Patents

Description

  The present invention relates to a method and an apparatus for calculating an optical band gap of a semiconductor material used when determining an optical band gap from a Tauc plot.

One index for evaluating the physical properties of semiconductor materials is an optical band gap (hereinafter referred to as “band gap” where appropriate). The band gap means the energy difference between the uppermost part of the valence band filled with electrons and the lowermost part of the conduction band in which no electrons exist. As described in Patent Document 1 and Non-Patent Document 1, the band gap can be obtained from the Tauc plot. The Tauc plot is based on the following relationship proposed by Tauc et al.
(hνα) ^{1 / n} = k (hν-Eg)
Here, h is the Planck constant, ν is the frequency, α is the extinction coefficient, k is the proportionality constant, and Eg is the band gap. n varies depending on the type of transition of the semiconductor material and is determined as follows.
For direct allowable transition: n = 1/2
For direct forbidden transition: n = 3/2
Indirect permissible transition: n = 2
Indirect forbidden transition: n = 3
For the sake of simplicity, it is assumed below that the transition of the semiconductor material is a direct allowable transition (that is, n = 1/2).

From the absorption spectrum where the vertical axis (y axis) is absorbance A and the horizontal axis (x axis) is wavelength λ, the vertical axis is the square of the product of energy hν and absorption coefficient α (hνα) ² , and the horizontal axis is energy hν. In addition, a Tauc plot can be obtained by converting each. The relationship between the absorbance A and the absorption coefficient α is defined by the following equation.
A = -ln (I / I ₀ ) = αL
Here, I is the intensity of light transmitted through the sample cell, I ₀ is the intensity of light incident on the sample cell, and L is the optical path length in the sample cell.

The above-described absorption coefficient α is a value specific to the substance and changes according to the energy of incident light. The absorption coefficient α of the semiconductor material increases as the wavelength λ of incident light becomes shorter, and increases greatly particularly in the vicinity of the wavelength corresponding to the band gap. Similarly, in the Tauc plot, the value of (hνα) ² increases little by little with the energy hν, greatly increases in the vicinity of the band gap, draws an S-shaped ascending curve, and then increases gently (FIG. 1 ( a)). When a tangent line passing through the inflection point of the S-shaped rising curve is drawn, the energy at the intersection of the tangent line and the x axis (y = 0 straight line) becomes the band gap Eg (see FIG. 1B). When a base line exists in the Tauc plot in addition to the x axis (y = 0 straight line), the intersection of the tangent line and the base line becomes the band gap Eg (see FIG. 1C).

Japanese Patent Laid-Open No. 11-14841

"Measurement of band gaps in compound semiconductors-Obtaining band gaps from diffuse reflectance spectra-", SHIMADZU APPLICATION NEWS, No. A428, October 2010

When obtaining the band gap Eg, first, in the region where the Tauc plot draws the S-shaped ascending curve, the primary differential value at each point of the Tauc plot is calculated, and the primary differential value is maximized. Find the point and determine the inflection point.
However, since noise at the time of measurement is superimposed on the absorption spectrum, the influence remains in the Tauc plot. For this reason, when the first derivative of the S-shaped curve in the Tauc plot is calculated, in many cases, a plurality of local maximum values appear. In some cases, the point where the first-order differential value is the maximum is not a true inflection point. Therefore, if the inflection point is automatically determined in the conventional apparatus, there is a possibility that an incorrect band is selected by selecting an incorrect point and creating a tangent equation.

  The problem to be solved by the present invention is to provide a method and a device for calculating an optical band gap of a semiconductor material, in which a correct band gap can be obtained from a curve on a Tauc plot created from measured data such as an absorption spectrum. It is to be.

The present invention made to solve the above problems is a calculation method used to determine the optical band gap of a semiconductor material from a Tauc plot,
a) calculating a primary differential value at each point of the Tauc plot, and calculating a primary differential value with X0 being an energy value at which the primary differential value is maximized;
b) For all combinations of two points separated by a predetermined number of plot points within a predetermined energy range including X0, an approximate line is obtained for Tauc plot points existing between the two points, and the approximate line An approximate straight line calculation step for obtaining the slope of
c) a band gap determination step in which the energy value of the intersection of the approximate line having the maximum inclination and the baseline of the Tauc plot among the approximate lines obtained in the approximate line calculation step is an optical band gap;
It is characterized by having.

The predetermined energy range including X0 is, for example, in consideration of the overall shape of the Tauc plot, considering that the inflection point exists in the energy range in which the Tauc plot draws an S-shaped rising curve. It is sufficient to set the energy range in which an S-shaped rising curve is drawn. The predetermined number of plot points may be set to be equal to or greater than the peak width of noise in consideration of the peak width of noise superimposed on the Tauc plot, for example. The approximate straight line can be obtained by, for example, the least square method.
The baseline of the Tauc plot means either a baseline calculated separately for the Tauc plot or an x axis (a straight line with y = 0). Therefore, in the band gap determination step, when the baseline of the Tauc plot is created in advance, from the intersection of the baseline and the approximate line, otherwise, the x axis (y = 0 straight line) and the approximate line The band gap is determined from the intersection point.

In the optical bandgap calculation method according to the present invention, in the approximate straight line calculation step, Tauc exists between the two points for all combinations of two points separated by a predetermined number of points within a predetermined energy range. The approximate straight line and the slope of the approximate straight line are comprehensively obtained for the plot points. Then, in the band gap determination step, an approximate line having the maximum slope is selected, and the band gap is determined from the intersection of the approximate line and the baseline of the Tauc plot.
As described above, in the optical band gap calculation method according to the present invention, an approximate straight line is comprehensively calculated, and an approximate straight line having the maximum inclination is selected from the approximate straight line to obtain a band gap. Can be sought.

The optical band gap calculation method according to the present invention is as follows.
d) Within the range from the Tauc plot point on the lowest energy side to the Tauc plot point that is smaller by a predetermined energy than X0, all the combinations of two points that are more than the predetermined number of plot points are A baseline determination step for determining an approximate line for Tauc plot points existing between points, and determining a baseline based on a predetermined condition from those approximate lines;
It is desirable to have

The predetermined energy is, for example, that the energy range of the Tauc plot used when obtaining the approximate line in the second approximate line calculation step is lower than the area where the Tauc plot draws an S-shaped ascending curve. What is necessary is just to set. Further, the predetermined number of points may be determined so that, for example, the range determined by the number of points is sufficiently wider than the peak width of each noise superimposed on the Tauc plot.
Also in the baseline determination step, the approximate straight line can be obtained by, for example, the least square method. In this case, the predetermined condition may be that the approximate straight line having the smallest square error is used as the baseline.

The optical bandgap calculation method according to the present invention further includes:
e) It is desirable to have a smoothing process step for smoothing the Tauc plot.
The smoothing process step is performed before the first derivative calculation step. Thereby, the influence of the noise superimposed on the Tauc plot can be reduced, and a correct band gap can be obtained with high accuracy. In this case, for example, smoothing differentiation using the Savitzky-Golay method may be performed, and the smoothing processing step and the primary differential value calculation step may be performed simultaneously.

Another aspect of the present invention made to solve the above problems is used for obtaining an optical band gap of a semiconductor material from a Tauc plot, and a parameter input means for allowing a user to input parameters necessary for the purpose. A calculation device comprising:
a) a primary differential value calculating means for calculating a primary differential value at each point of the Tauc plot and setting the energy value at which the primary differential value is maximum as X0;
b) Based on the above parameters, for all combinations of two points separated by a predetermined number of plot points within a predetermined energy range including X0, an approximate straight line targeting Tauc plot points existing between the two points And an approximate straight line calculating means for determining the slope of the approximate straight line;
c) Among the approximate lines obtained by the approximate line calculation means, an approximate straight line having the maximum inclination, and a band gap determination means having an optical band gap as an energy value at the intersection of the base lines of the Tauc plot;
It is characterized by providing.

  The predetermined energy range and the predetermined number of plot points are included in the parameters input by the user using the parameter input means. As described above, the user sets the predetermined energy range in consideration of the shape of the Tauc plot, for example, and sets the predetermined energy range in consideration of the peak width of the noise superimposed on the Tauc plot, for example. Set a fixed number of points.

In addition, the optical bandgap calculation apparatus according to the present invention includes:
d) Based on the above parameters, a combination of two points separated from the Tauc plot point on the lowest energy side by a predetermined number of plot points within the range from the Tauc plot point that is smaller than X0 by a predetermined energy. For all of the above, obtaining an approximate line for Tauc plot points existing between the two points, and based on the parameters, a baseline determining means for determining a baseline from among the approximate lines;
It is desirable to have

  In the case of the apparatus of the aspect provided with the baseline determination means, the parameter input means allows the user to input parameters necessary for processing to be performed by the baseline determination means.

The optical bandgap calculation apparatus according to the present invention further includes:
e) It is desirable to have smoothing processing means for smoothing the Tauc plot.

In the optical bandgap calculation method according to the present invention, an approximate straight line is comprehensively calculated, and an approximate straight line having the maximum inclination is selected from the approximate straight line. The correct band gap can be obtained from the obtained Tauc plot.
Similarly, when the optical bandgap calculation apparatus according to the present invention is used, a correct bandgap can be obtained from a Tauc plot created from measured data.

The figure explaining the method of calculating a band gap from Tauc plot. The principal part block diagram of the calculation apparatus of the band gap of the semiconductor substance of a present Example. 6A and 6B illustrate a procedure for determining a tangent and a band gap in a method for calculating a band gap of a semiconductor material according to an embodiment. 6A and 6B illustrate a procedure for determining a base line and a band gap in a method for calculating a band gap of a semiconductor material according to an embodiment. The example which calculated the band gap from the Tauc plot using the calculation method and calculation apparatus of the band gap of the semiconductor substance of a present Example.

The Tauc plot can be obtained from the absorption spectrum as described above, and can also be obtained from the diffuse reflection spectrum. For example, when the sample is powder, a Tauc plot is obtained from the diffuse reflection spectrum. In this embodiment, the Tauc plot is obtained from the diffuse reflection spectrum, and the band gap is calculated. When obtaining a Tauc plot from a diffuse reflection spectrum, first, the diffuse reflection spectrum is subjected to Kubelka-Munk conversion. Specifically, the Kubelka-Munk transformation is a method of converting the vertical axis of the diffuse reflection spectrum into an amount F (R _∞ ) proportional to the absorption coefficient α, and is defined by the following equation.
F (R _∞ ) = (1-R _∞ ) ² / 2R _∞ = α / S
Here, _R∞ is the absolute reflectance, and S is the scattering coefficient.
Subsequently, the vertical axis (F (R _∞ )) and horizontal axis (wavelength nm) of the Kubelka-Munk transformed spectrum are converted into the vertical axis ((hνF (R _∞ )) ² ) and horizontal axis (hν), respectively. . As described above, since F (R _∞ ) is proportional to the absorption coefficient α, the band gap can be calculated by the procedure described above. In the present embodiment, those having the vertical axis and the horizontal axis are also called Tauc plots.

  An embodiment of a method and apparatus for calculating an optical band gap of a semiconductor material according to the present invention (hereinafter simply referred to as “calculation method” and “calculation apparatus”) will be described with reference to the drawings. FIG. 2 is a block diagram of the main part of the calculation apparatus of the present embodiment, and FIGS.

  The calculation apparatus according to the present embodiment includes a storage unit 10, a parameter input unit 11, a smoothing processing unit 12, a primary differential value calculation unit 13, an approximate straight line calculation unit 14, a baseline determination unit 15, and a band gap determination unit 16. A computer having a function, and an input unit 20 and a display unit 30 connected to the computer are provided. The operation of each unit will be described together with the calculation method according to the present embodiment.

First, the Tauc plot obtained by the above-described procedure is read into a computer and stored in the storage unit 10 (step S1). Subsequently, the parameter input unit 11 displays a parameter input screen on the display unit 30 and allows the user to input the following parameters 1 to 6. Details of each parameter will be described in detail in the step in which the parameter is used. A numerical value or the like described in parentheses of each parameter is a specific value or the like input by the user in this embodiment.
Parameter 1: Data interpolation interval setting for Tauc plot in step S2 (0.001eV)
Parameter 2: Presence / absence of smoothing process in step S4 and its method (with smoothing process + moving average)
Parameter 3: Target energy range in step S6 (± 0.1 eV)
Parameter 4: 2 points selected in step S6 (10 points)
Parameter 5: Energy range targeted in step S21 (-0.1 eV)
Parameter 6: Conditions for two points selected in step S21 (50 points)

The smoothing processing unit 12 checks whether the plot interval of the Tauc plot stored in the storage unit 10 matches the parameter 1 (0.001 eV) input by the user (step S2). If not, the Tauc plot is complemented (interpolated) so that the plot interval of the Tauc plot matches the parameter 1 (0.001 eV) (step S3).
Usually, when measuring the diffuse reflection spectrum, the wavelength (λ) is changed by a certain value. In the Tauc plot, since the horizontal axis is converted to energy (hν), the plot intervals of the Tauc plot after the Kubelka-Munk conversion of the diffuse reflection spectrum are uneven. Therefore, in the present embodiment, the Tauc plot interval is complemented (interpolated) at equal intervals. The value of parameter 1 is set in consideration of the number of significant digits (three digits after the decimal point in this embodiment) necessary for calculating the band gap.

  Further, the smoothing processing unit 12 smoothes the Tauc plot based on the parameter 2 (with smoothing processing + moving average) (step S4). As parameter 2 (smoothing method), various other methods such as smoothing differentiation using the Savitzky-Golay method can be used. In this case, the first-order differential value calculation described later is performed simultaneously. Further, when the influence of noise superimposed at the time of measurement is small and a smooth Tauc plot is obtained, the parameter 2 may be set as no smoothing processing.

  The primary differential value calculation unit 13 calculates the primary differential value at each point of the Tauc plot, and sets the energy value at which the primary differential value is maximum as X0 (step S5).

  Next, the approximate straight line calculation unit 14 is based on the parameter 3 (± 0.1 eV) and the parameter 4 (10 points). And an approximate straight line is obtained by the least square method for all Tauc plot points between the two points (step S6). Then, the obtained inclination A of the approximate straight line is stored as Amax in the storage unit 10 (step S7).

Parameter 3 is set by confirming the range in which the Tauc plot draws an S-shaped ascending curve. In this embodiment, the parameter 3 is set to ± 0.1 eV. For example, when the S-shaped rising curve is steep, it can be set to ± 0.05 eV, or can be set to ± 0.2 eV when the curve is gentle. .
Parameter 4 is set so that at least three measured points of the diffuse reflection spectrum are included between the two extracted points in order to reduce the influence of noise superimposed on the Tauc plot. When acquiring a diffuse reflection spectrum of a semiconductor material, for example, the wavelength is changed by 0.1 nm in a range of 200 nm to 1600 nm. The shortest wavelength in this wavelength range is about 200 nm. If the 0.1 nm wavelength is different in this wavelength range, the energy is about 0.003 eV different. In consideration of this point, the parameter 4 is set to 10 points (corresponding to 0.01 eV) in this embodiment.
The setting of the parameter 3 and the parameter 4 is not limited to these, and may be set in consideration of the time required from step S6 to step S10 described below, for example.

  Subsequently, it is confirmed whether the slopes of the approximate lines have been compared for all combinations of two points separated by 10 points or more in the energy range from X0-0.1 eV to X0 + 0.1 eV (step S10). In the first stage, since the comparison of the slopes of the approximate straight lines for all the combinations has not been completed, the process returns to step S6 again to extract two points that are 10 points or more apart from the two previously extracted points. Similarly, an approximate straight line is obtained. Then, the inclination A of the approximate line is compared with Amax stored in the storage unit (step S7). If A> Amax (step S8), the newly obtained inclination A of the approximate line is calculated. The new Amax is stored in the storage unit 10 (step S9). If A <Amax in step S8, the previously stored Amax is stored as it is. When the comparison of the slopes of the approximate lines for all the combinations is completed, the approximate line having the slope of Amax stored in the storage unit 10 is determined as the tangent line of the Tauc plot (step 11). Finally, the band gap determining unit 16 determines the energy value Eg at the intersection of the tangent line of the Tauc plot and the x axis (y = 0 straight line) as the band gap (step 12). If there is a baseline in the Tauc plot that is separate from the x axis (y = 0 line), the band gap determination unit 16 calculates the baseline through the procedure described below and then determines the band gap. To do.

  A base line calculation procedure in the case where a base line exists in the Tauc plot separately from the x-axis (y = 0 straight line) will be described with reference to FIGS. In the present embodiment, steps S21 to S29 relating to the baseline calculation are performed instead of step S12 after step S11 is completed. Note that steps S21 to S29 relating to the baseline calculation can be executed after step S5 described above is completed, that is, after X0 is determined. Therefore, for example, steps S21 to S29 may be executed in parallel with the above-described steps S5 to S11.

  The baseline determination unit 15 is based on the parameter 5 (−0.1 eV) and the parameter 6 (50 points), and is an arbitrary 2 separated by 50 points or more in the energy range of X0-0.1 eV from the Tauc plot point on the lowest energy side. Points are extracted (step S21), and an approximate straight line is obtained by the least square method for all Tauc plot points between the two points (step S22). Then, a square error E between the approximate straight line and each plot point of the Tauc plot is calculated (step S23), and is stored in the storage unit 10 as Emin (step S24).

Parameter 5 is set so that the range in which the Tauc plot draws an S-shaped rising curve is confirmed, and the lower energy side of the energy range where the rising curve is located is the target range. In this embodiment, the parameter 5 is set to -0.1 eV. For example, when the S-shaped rising curve is steeper, it is set to -0.05 eV, and when it is gentle, it is set to -0.2 eV. Can do.
Parameter 6 can be set in the same way as parameter 4 described above. However, since the target energy range in step S21 to step S26 to which parameter 6 is applied is wider than the target energy range in step S6 to step S10 to which parameter 4 is applied, the number of combinations of two points to be extracted Increases, and processing takes time. Therefore, the time required for performing steps S21 to S26 is set taking into account simultaneously. That is, when the energy range is narrow, the number of parameters 6 is set small, and when the energy range is wide, the number of parameters 6 is set large.
These parameter setting methods are merely examples. For example, in consideration of the overall shape of the Tauc plot, an energy range in which the linearity of the Tauc plot is high on the low energy side may be selected. Thereby, a baseline can be determined with high accuracy.

  Subsequently, it is confirmed whether the square error of the approximation line has been compared for all combinations of two points separated by 50 points or more in the energy range of X0-0.1 eV from the Tauc plot point on the lowest energy side (step S27). ). In the first stage, since the comparison of the square errors of the approximate straight lines for all the combinations has not been completed, the process returns to step S21 again, and two points that are 50 points or more apart from the two previously extracted points are extracted. Similarly, an approximate straight line is obtained (step S22). Then, the square error E of this approximate straight line is calculated (step S23), the square error E is compared with Emin stored in the storage unit 10 (step S24), and if E <Emin (step S25). In this case, the square error E of the newly obtained approximate straight line is stored in the storage unit 10 as a new Emin (step S26). If E> Emin in step S24, the previously stored Emin is stored as it is. When the comparison of the slopes of the approximate lines for all the combinations is completed, the approximate line corresponding to the square error Emin stored in the storage unit 10 is determined as the baseline of the Tauc plot (step 28). Finally, the band gap determination unit 16 determines the energy value Eg at the intersection of the tangent line of the Tauc plot and the baseline as the band gap (step S29).

FIG. 5 shows an example in which the band gap is calculated from the Tauc plot by the calculation device and the calculation method of this example. In this calculation example, the Tauc plot is smoothed and differentiated using the Savitzky-Golay method instead of the smoothing process based on the moving average. FIG. 5 (a) shows CuInSe ₂ , FIG. 5 (b) shows CuIn _0.5 Ga _0.5 , and FIG. 5 (c) shows CuGaSe ₂ calculated band gaps from Tauc plots obtained from diffuse reflection spectra. As can be seen from these results, even when the shape of Tauc plot is different or the influence of noise remains in Tauc plot, the true inflection point is selected by using the calculation device and calculation method of this embodiment. Thus, a correct band gap can be obtained.

The above-described embodiment is an example, and can be appropriately changed in accordance with the gist of the present invention. In particular, any of the parameters used in the above embodiment is an example, and is appropriately changed in consideration of the shape of the Tauc plot.
In steps S6 and S22 of the above embodiment, approximate straight lines are determined for all the Tauc plot points existing between the two extracted points. However, in order to shorten the calculation time, Tauc plots positioned at predetermined intervals are used. Points may be extracted, and an approximate straight line may be determined for only those plot points. In the present embodiment, the Tauc plot interval is complemented (interpolated) at equal intervals, but the band gap can be calculated without performing this processing.

DESCRIPTION OF SYMBOLS 10 ... Memory | storage part 11 ... Parameter input part 12 ... Smoothing process part 13 ... Primary differential value calculation part 14 ... Approximate straight line calculation part 15 ... Baseline determination part 16 ... Band gap determination part 20 ... Input part 30 ... Display part

Claims (10)

a) calculating a primary differential value at each point of the Tauc plot, and calculating a primary differential value with X0 being an energy value at which the primary differential value is maximized;
b) For all combinations of two points separated by a predetermined number of plot points within a predetermined energy range including X0, an approximate line is obtained for Tauc plot points existing between the two points, and the approximate line An approximate straight line calculation step for obtaining the slope of
c) a band gap determination step in which the energy value of the intersection of the approximate line having the maximum inclination and the baseline of the Tauc plot among the approximate lines obtained in the approximate line calculation step is an optical band gap;
A method for calculating a band gap of a semiconductor material, comprising: 2. The method of calculating a band gap of a semiconductor material according to claim 1, wherein, in the approximate line calculation step, an approximate line is obtained using a least square method. d) Within the range from the Tauc plot point on the lowest energy side to the Tauc plot point that is smaller by a predetermined energy than X0, all the combinations of two points that are more than the predetermined number of plot points are determine the approximate straight line as target Tauc plot points present between the points, the claims from these approximate lines, and having a baseline determination step of determining a baseline based on a predetermined condition A method for calculating a band gap of the semiconductor material according to 1 or 2 . Prior Symbol baseline determination step, the method for calculating the band gap of the semiconductor material according to claim 3, wherein the determination of the approximate straight line using the least squares method. 5. The method of calculating a band gap of a semiconductor material according to claim 1, further comprising a smoothing step for performing a smoothing process on e) Tauc plot. A calculation device having a parameter input means that is used to obtain an optical band gap of a semiconductor material from a Tauc plot and allows a user to input parameters necessary for the purpose.
a) a primary differential value calculating means for calculating a primary differential value at each point of the Tauc plot and setting the energy value at which the primary differential value is maximum as X0;
b) Based on the above parameters, for all combinations of two points separated by a predetermined number of plot points within a predetermined energy range including X0, an approximate straight line targeting Tauc plot points existing between the two points And an approximate straight line calculating means for determining the slope of the approximate straight line;
c) Among the approximate lines obtained by the approximate line calculation means, an approximate straight line having the maximum inclination, and a band gap determination means having an optical band gap as an energy value at the intersection of the base lines of the Tauc plot;
An apparatus for calculating a band gap of a semiconductor material, comprising: The apparatus for calculating a band gap of a semiconductor material according to claim 6, wherein the approximate line calculation means obtains an approximate line using a least square method. d) Based on the above parameters, a combination of two points separated from the Tauc plot point on the lowest energy side by a predetermined number of plot points within the range from the Tauc plot point that is smaller than X0 by a predetermined energy. Characterized in that it has a baseline determining means for determining an approximate line for Tauc plot points existing between the two points and determining a baseline from the approximate lines based on the parameters. The apparatus for calculating a band gap of a semiconductor material according to claim 6 or 7 . Before SL baseline determining means, calculating device of the band gap of the semiconductor material according to claim 8, wherein the determination of the approximate straight line using the least squares method. The apparatus for calculating a band gap of a semiconductor material according to claim 6, further comprising: e) a smoothing processing unit that performs Tauc plot smoothing processing.

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