A method for determining the thermodynamic temperature of a sample surface based on photoelectron spectroscopy.

Abstract

To provide a method for determining a thermodynamic temperature of a sample surface based on a photoelectron spectrum, in which temperature accuracy is improved and which is suitable for practical temperature measurement.SOLUTION: In a standardized photoelectron spectrum of a sample surface, when an area ratio of an area SL at RL on a negative side of an energy range R to an area SU at RU on a positive side is defined as an area ratio rex, a method for determining a thermodynamic temperature of a sample surface based on a photoelectron spectrum determines a thermodynamic temperature T and a broadening width ΔE of a sample surface simultaneously by calculating the plurality of area ratios rex while changing a width of R, and fitting an area ratio function rth(EC) calculated using Equation (2) from a function FDG of Equation (1) to the plurality of area ratios rex.SELECTED DRAWING: Figure 7

Description

[Technical Field] 【0001】 The present invention relates to a method for determining the thermodynamic temperature of a sample surface based on a photoelectron spectrum, and more particularly, to a method for determining the thermodynamic temperature of a sample surface based on a Fermi-Dirac distribution obtained by measuring the electron energy distribution near the Fermi level for the sample surface. [Background technology] 【0002】 Methods for measuring the temperature of a sample surface are broadly classified into contact methods and non-contact methods. Contact methods are performed using temperature measuring devices such as thermocouples and resistance thermometers. Non-contact methods are performed using temperature measuring devices such as infrared thermometers. 【0003】 Thermocouples and resistance thermometers used in contact methods are widely used in various technical fields due to their simple device configuration. However, due to the large size of the sensor part of the device and the fact that the sensor part comes into contact with the sample surface, they are not an effective measurement method for clean surfaces, thin film surfaces in thin film manufacturing processes, or sample surfaces controlled on the nanometer scale. In recent years, attempts have been made to perform localized temperature measurements using scanning thermal microscopes (SThMs), which have thermocouples or resistance thermometers attached to the tip of an atomic force microscope (AFM). However, the physical quantities measured thermally by SThMs have ambiguous definitions, and there are problems with the quantitative accuracy and reproducibility of temperature measurements. Furthermore, because SThMs have a complex device configuration, simultaneous multi-functional measurements with other surface analysis measurements are not possible. 【0004】 Infrared thermometers used in non-contact thermometering methods measure thermal radiation from the sample surface and determine the sample surface temperature from its wavelength distribution and intensity. Infrared thermometers are widely used as practical thermometers. However, since real materials are not ideal blackbody radiation sources, the amount of thermal radiation from the sample surface depends on the thermo-optic properties (emissivity) of that surface. Therefore, without detailed information about the emissivity of the material constituting the sample, it is impossible to accurately determine the sample surface temperature. 【0005】 In light of the problems described above, there is a need for a practical temperature measurement method that satisfies all three conditions of measurement needs in the fields of surface analysis and nanomeasurement: locality (surface selectivity), non-contact, and simultaneous multi-functional measurement. 【0006】 Incidentally, the occupancy rate of electronic states in a substance at thermal equilibrium is described as a function of thermodynamic temperature by the Fermi-Dirac (FD) distribution. The FD distribution is also called the Fermi distribution function and is defined by the following equation. In the following equation, T is the thermodynamic temperature, and E F is the Fermi level, k B represents the Boltzmann constant. The function f(E,T) represents the probability that an electronic state of a certain energy E is occupied in an electron gas in thermal equilibrium at thermodynamic temperature T. 【0007】 【number】 【0008】 The FD distribution is a function that depends only on the thermodynamic temperature T. The inventors have found that if a method is developed to determine the thermodynamic temperature based on the FD distribution obtained by measuring the electron energy distribution near the Fermi level on the sample surface with high resolution, it will be possible to measure the temperature that satisfies the three conditions mentioned above. 【0009】 Conventionally, scanning tunneling spectroscopy (STM) and ultraviolet photoelectron spectroscopy (UPS) are known methods for measuring the electronic state of a sample surface. STM is suitable for measuring the local density of electronic states on a sample surface. However, the intensity of the tunneling current obtained by the measurement is the integral of the electronic state of the sample surface and the electronic state of the wire-like metal, and it is not possible to directly measure the electron energy distribution near the Fermi level on the sample surface. UPS is a method that irradiates the sample surface with monochromatic light in the ultraviolet or vacuum ultraviolet region and measures the kinetic energy of excited electrons emitted from the sample surface. In UPS, the escape depth of excited electrons from the sample is several atomic layers, and it measures the energy distribution of electrons localized on the sample surface. The high-energy edge (Fermi edge) of the photoelectron spectrum measured by UPS is the photoelectron emission from the Fermi level on the sample surface, and the photoelectron intensity at the start of the photoelectron spectrum reflects the FD distribution of the sample. Temperature measurement of the sample surface by UPS is described, for example, in Non-Patent Documents 1 and 2. However, while the temperature resolution (temperature accuracy) required for practical temperature measurement is less than 1K, the temperature accuracy described in Non-Patent Documents 1 and 2 is much higher, at several tens of K, making it impractical. 【0010】 The inventors proposed an electron energy analyzer specifically for measuring thermodynamic temperature (Patent Document 1). Furthermore, the inventors used an existing photoelectron spectrometer to measure the electron energy distribution of a Cu(110) surface in the temperature range of 10 to 300 K. This revealed that the thermodynamic temperature T cannot be accurately determined using only the FD distribution, and that to determine the thermodynamic temperature T more accurately, it is necessary to determine the broadening width ΔE of the photoelectron spectrum simultaneously with the thermodynamic temperature T from the photoelectron spectrum and consider this (Non-Patent Document 3 by the inventors). Subsequently, the inventors performed high-energy-resolution photoelectron spectroscopy measurements of an Au(110) surface at a temperature of around 100 K. This allowed us to simultaneously determine the thermodynamic temperature T and the broadening width ΔE by fitting a function obtained by linearly combining a Gaussian function and a Lorentz function as the broadening function, and we found that the thermodynamic temperature T determined in this way had an accuracy of 99 ± 2.1 K relative to the temperature value of 98.5 ± 0.5 K measured by a temperature sensor (Non-Patent Literature 4 by the present inventors). Furthermore, in order to discuss the broadening function in detail, the present inventors found that the broadening function could be derived from the measured photoelectron spectrum using the Fourier transform (Non-Patent Literature 5 by the present inventors). The present inventors then found that by eliminating the intensity change due to the density of states (DOS) of electrons from the photoelectron spectrum, the Gaussian function could be treated as the broadening function. [Prior art documents] [Patent Documents] 【0011】 [Patent Document 1] Patent No. 5545577 [Non-patent literature] 【0012】 [Non-Patent Document 1] SS Mann, BD Todd, JT Stuckless, T. Seto, and DA King, Pulsed laser surface heating: nanosecond time-scale temperature measurement, Chem. Phys. Let. 183, 529 (1991) [Non-Patent Document 2] J. Kroger, T. Greber, TJ Ktrutz, and J. Osterwalder, The photoemission Fermi edge as a sample thermometer?, J. Electr. Spec. Rel. Phenom. 113, 241 (2001) [Non-Patent Document 3] I. Kinoshita and J. Ishii, Preliminary Experiments of Photoelectron Thermometry, Temperature: Its Measurement and Control in Science and Industry, 8, 915 (2013) [Non-Patent Document 4] I. Kinoshita, C. Tsukada, K. Ouchi, E. Kobayashi, and J. Ishii, A method for atomic-level noncontact thermometry with electron energy distribution, Jap. J. Appl. Phys. 56, 048004 (2017) [Non-Patent Document 5] T. Nishida, I. Kinoshita, J. Ishii, Temperature-Dependent Broadening of the Ultraviolet Photoelectron Spectrum of Au(110), Sensors 21, 5969 (2021) [Overview of the project] [Problems that the invention aims to solve] 【0013】 Based on the above findings, the inventors have discovered a method for determining the thermodynamic temperature of a sample surface based on the photoelectron spectrum. This method involves fitting a function (equation (1) described later) obtained by convolving and integrating a Gaussian function with broadening width ΔE as the broadening function with respect to a FD distribution with thermodynamic temperature T as the variable, to the photoelectron spectrum near the Fermi level of the sample surface, thereby simultaneously determining the thermodynamic temperature T and broadening width ΔE, which are fitting variables. 【0014】 However, when the above method is applied to photoelectron spectra measured with a sufficient number of integrations using existing high-resolution electron energy analyzers, the temperature accuracy is only about 2-3 K, which falls short of the 1 K benchmark required for practical temperature measurement. 【0015】 One method to obtain sufficient temperature accuracy and reliability for practical temperature measurement is to improve the signal-to-noise ratio (S / N ratio) of the photoelectron spectrum. The S / N ratio increases as the number of integrations of the photoelectron spectrum increases, and the photoelectron spectrum becomes more averaged. However, increasing the number of integrations to improve the S / N ratio of the photoelectron spectrum can increase the measurement time, which may cause the temperature of the sample surface to change. Furthermore, there are limits to the averaging of the photoelectron spectrum by increasing the number of integrations, and it is impossible to completely eliminate measurement noise. Another method is to remove noise using a bandpass filter, but smoothing the waveform of the photoelectron spectrum with a bandpass filter may also change the shape of the electron energy distribution. If the shape of the electron energy distribution changes, the temperature value determined will also change. 【0016】 This invention was made based on the circumstances described above, and aims to provide a method for determining the thermodynamic temperature of a sample surface based on a photoelectron spectrum, which has improved temperature accuracy and is suitable for practical temperature measurement. 【Means for Solving the Problem】 【0017】 The present invention provides a method for determining the thermodynamic temperature by using the area of a photoelectron spectrum. For a photoelectron spectrum with measurement noise, by using the area ratio of the spectrum rather than the intensity of the spectrum as the fitting target, the influence of the measurement noise can be suppressed without losing the information on the electron energy distribution of the photoelectron spectrum. Thereby, the thermodynamic temperature can be determined with high accuracy and reliability. 【0018】 To solve the above problems, first, the present invention is a method for determining the thermodynamic temperature of a sample surface based on a photoelectron spectrum, comprising measuring the photoelectron spectrum of the sample surface, eliminating the intensity change caused by elements other than the Fermi-Dirac (FD) distribution from the measured photoelectron spectrum, and further normalizing it. In the normalized photoelectron spectrum, taking the Fermi level as the reference zero point, the positive-side energy E in the axial direction of the electron energy E C and the negative-side energy -E C a pair of ranges (-E C < E < E C ) are defined as R. Among R, the area of the photoelectron spectrum in the positive-side range (0 < E < E C ) is defined as S U . Among R, the area of the photoelectron spectrum in the negative-side range (-E U < E < 0) is defined as S C . When the ratio S L to S L is defined as the area ratio r L / S U and denoted as r U / S L , while changing the energy E ex and changing the width of R, a plurality of area ratios r C are calculated. For the calculated plurality of area ratios r ex , from the function FDG obtained by convolution integration of the Gaussian function with the FD distribution function represented by the following formula (1), the area ratio function r ex calculated using the following formula (2) th (EC By fitting (1), the thermodynamic temperature T of the sample surface and the broadening width ΔE are simultaneously determined, providing a method including this (Invention 1). 【0019】 【Number】 【0020】 Here, the function FDG in the above formula (1) is a function obtained by convolving and integrating a Gaussian function with the broadening width ΔE as a variable with respect to the FD distribution function having the thermodynamic temperature T as a variable. In the above formula (1), E is the electron energy, E F is the Fermi level, k B is the Boltzmann constant, and ε represents the intermediate variable in the convolution integral. 【0021】 【Number】 【0022】 Here, the area ratio function r th (E C ) is a function of the variable E that represents the ratio of the area of the spectrum in the range above the Fermi level (0 < E < E C < E < 0) to the area of the spectrum in the range below the Fermi level (-E C ), using the function that gives the theoretically obtained spectral intensity in the above formula (1). C is a function of. 【0023】 According to such an invention (Invention 1), a method for determining the thermodynamic temperature of a sample surface based on a photoelectron spectrum, which has improved temperature accuracy and is suitable for practical temperature measurement, can be provided. 【0024】 In the above invention (Invention 1), the intensity change caused by factors other than the above FD distribution may be the intensity change due to the density of states (DOS) of electrons (Invention 2). 【0025】 In the above inventions (Inventions 1 and 2), it is preferable to normalize the photoelectron spectrum using the following formula (3) (Invention 3). 【0026】 【number】 【0027】 Here, in equation (3) above, I represents the intensity of the photoelectron spectrum, A represents the amplitude, and B represents the base. Typically, f can be the FD distribution function. 【0028】 In the above invention (inventions 1-3), multiple area ratios r ex To calculate the energy E, the standard deviation σ obtained by differentiating the normalized photoelectron spectrum with respect to energy and fitting it to a Gaussian function is used. C Determine the upper limit of the area ratio r ex This includes calculating (Invention 4). 【0029】 The above inventions (Inventions 1-4) preferably further include determining the thermodynamic temperature T using the least squares method (Invention 5). 【0030】 In the above invention (Invention 5), χ is defined by the following formula (4). 2 The deviation Δχ is calculated using the following formula (5). 2 It is preferable to determine the temperature accuracy dT of the thermodynamic temperature T based on the above (Invention 6). 【0031】 【number】 【0032】 Here, in equation (4) above, n represents the number of data points and m represents the number of parameters. T^ is the optimal value of T determined by fitting, S(T) represents the least squares sum as a function of T, with the three fitting parameters other than T, calculated by the least squares method, fixed to their optimal values. S0(T^) represents the least squares sum with all four parameters fixed to their optimal values. 【0033】 【number】 【0034】 Here, Δχ in equation (5) above 2 χ is calculated as a function of T. 2 This represents the deviation. 【0035】 In the above inventions (Inventions 1-6), it is preferable that the sample is in a temperature range below the melting point of the sample (Invention 7). [Effects of the Invention] 【0036】 According to the present invention, a method for determining the thermodynamic temperature of a sample surface based on a photoelectron spectrum can be provided, which offers improved temperature accuracy and is suitable for practical temperature measurement. [Brief explanation of the drawing] 【0037】 [Figure 1] This image shows the overall photoelectron spectrum of an Au(110) single crystal surface relative to the Fermi level, and a magnified view of the area near the Fermi level. [Figure 2] This figure shows the photoelectron spectrum near the Fermi level on the surface of an Au(110) single crystal. [Figure 3] This is a normalized version of the photoelectron spectrum shown in Figure 2. [Figure 4] This figure shows a portion of the band curve (Ek curve) representing the electronic band structure of an Au(110) single crystal, as determined by first-principles electronic structure calculations (band calculations). [Figure 5] Figure 4 shows the density of states (DOS) of electrons obtained by counting the number of states at 10 meV intervals in the band curve. [Figure 6] This diagram illustrates the procedure for eliminating changes in the photoelectron spectrum due to DOS of electrons and for normalization using equation (3). [Figure 7]This diagram illustrates the range R, positive range RU, negative range RL, area SU, and area SL in relation to the energy EC. [Figure 8] This diagram illustrates how to determine the upper limit of EC when calculating the area ratio rex. [Figure 9] This graph plots multiple area ratios (rex) calculated by varying the EC (electron emission coefficient) from a normalized photoelectron spectrum. [Figure 10] This graph shows an example of the area ratio function rth(EC), calculated when ΔE is assumed to be 30 meV and T is set to 50K, 100K, and 300K. [Figure 11] This figure shows the relationship between the deviation Δχ² and the probability distribution of the deviation ΔT from the optimal temperature value T determined by fitting. [Figure 12A] This figure shows the results of fitting the area ratio function rth(EC) using the least squares method to multiple area ratios rex calculated while varying EC for Example 1. [Figure 12B] This figure shows the relationship between the deviation Δχ² and the fitting parameter T for Example 1. [Figure 13A] This figure shows the results of fitting the area ratio function rth(EC) using the least squares method to multiple area ratios rex calculated while varying EC in Example 2. [Figure 13B] This figure shows the relationship between the deviation Δχ² and the fitting parameter T for Example 2. [Figure 14A] This figure shows the results of fitting the area ratio function rth(EC) using the least squares method to multiple area ratios rex calculated while varying EC in Example 3. [Figure 14B] This figure shows the relationship between the deviation Δχ² and the fitting parameter T for Example 3. [Modes for carrying out the invention] 【0038】 Hereinafter, embodiments of the method for determining the thermodynamic temperature of a sample surface based on the photoelectron spectrum of the present invention will be described with reference to the drawings as appropriate. The embodiments described below are provided to facilitate understanding of the present invention and do not limit the present invention in any way. 【0039】 Photoelectron spectra reflect the electron density (DOS) and reveal the electronic structure of a sample. Figure 1 is an overall view of the photoelectron spectrum measured perpendicular to the surface of an Au(110) single crystal, with the Fermi level as the reference. Measurements perpendicular to the surface (110) measure the electronic state in the wavenumber Γ-K direction of the electronic structure. In Figure 1, the horizontal axis represents electron energy (eV), and the vertical axis represents photoelectron intensity (arbitrary units). As shown in Figure 1, the photoelectron spectrum perpendicular to the surface of an Au(110) single crystal shows a d-band with high electron density between -2eV and -8eV, and an s / p-band with low electron density that is widely distributed in the spectrum. Furthermore, a step-like edge is observed near zero electron energy, i.e., near the Fermi level. An enlarged spectrum near the Fermi level is inserted in Figure 1. The inset in Figure 1 corresponds to Figure 2, which will be described later. These characteristics are also observed in the photoelectron spectra of noble metals other than Au, and represent the electron occupancy in the DOS. Band calculations show that the electron DOS in the wavenumber Γ-K direction near the Fermi level is almost constant. The step-like edge shape of the photoelectron spectrum is mainly created by the FD distribution, and this step-like function strongly depends on the thermodynamic temperature T of the sample surface. For this reason, UPS can be used as a method for measuring thermodynamic temperature. 【0040】 The method for determining the thermodynamic temperature of a sample surface based on the photoelectron spectrum according to this embodiment involves measuring the photoelectron spectrum of the sample surface (Step 1), eliminating intensity changes caused by elements other than the FD distribution from the measured photoelectron spectrum and further normalizing it (Step 2), and, in the normalized photoelectron spectrum, using the Fermi level as the reference zero, determining a certain energy E CIn contrast, a pair of ranges (-E) are formed on the positive and negative sides in the axial direction of the electron energy E. C <E<E C Let R be defined as the positive range of R (0 <E<E C )R U The area in S U Defined as the negative range of R (-E C <E<0)R L The area in S L Defined as, S L S for U Ratio S U / S L area ratio r ex When defined as, energy E C By changing the width of R, multiple area ratios r ex Calculate (Step 3) and the multiple area ratios r ex For this, the area ratio function r is calculated using equation (2) below from the function FDG obtained by convolving and integrating the FD distribution function with a Gaussian function, which is represented by equation (1) below. th (E C This includes simultaneously determining the thermodynamic temperature T and broadening width ΔE of the sample surface by fitting the sample (step 4). 【0041】 【number】 【0042】 Here, the function FDG in equation (1) above is the function FDG(E;T,ΔE) obtained by convolving and integrating an FD distribution with thermodynamic temperature T as a variable with a Gaussian function whose broadening function is the spectral spread, i.e., broadening width ΔE. In equation (1) above, E is the electron energy, E F is the Fermi level, k B ε represents the Boltzmann constant, and ε represents the parameter of the convolution integral. 【0043】 【number】 【0044】 Here, the area ratio function r of the above formula (2) th is a function that represents the ratio of the area of the spectrum in the range below the Fermi level (-E C < E < 0) to the area of the spectrum in the range above the Fermi level (0 < E < E C ) with respect to the spectrum area in the range below the Fermi level, using a function that gives the theoretically obtained spectral intensity from the above formula (1). C It is a function of the variable E. 【0045】 Examples of the sample to which this method is applied include metals or semiconductors such as Au(110) and Cu(110). 【0046】 This method is effective for knowing the thermodynamic temperature T of the sample surface in the temperature range where UPS can be carried out, that is, the temperature range where the temperature of the sample is below the melting point of the sample. For example, by this method, the thermodynamic temperature T of the sample surface under liquid nitrogen cooling temperature, under liquid helium cooling temperature, and at room temperature can be accurately determined. 【0047】 Hereinafter, this method will be described in detail with reference to FIGS. 2 to 11. 【0048】 (Step 1) In Step 1, the photoelectron spectrum of the sample surface is measured. In this embodiment, the photoelectron spectrum means the energy distribution of electrons obtained by ultraviolet photoelectron spectroscopy (UPS). FIG. 2 is a diagram showing the photoelectron spectrum of the Au(110) single crystal surface. In FIG. 2, the horizontal axis represents the kinetic energy (eV) of the measured electrons, and the vertical axis represents the photoelectron intensity (arbitrary unit). 【0049】 (Step 2) In Step 2, the intensity change caused by factors other than the FD distribution is eliminated from the photoelectron spectrum measured in Step 1, and further normalized. FIG. 3 is a diagram showing the normalization of the photoelectron spectrum of FIG. 2. In FIG. 3, the horizontal axis represents the electron energy (eV) based on the Fermi level (E F = 0), and the vertical axis represents the normalized photoelectron intensity (arbitrary unit). 【0050】 In this embodiment, the intensity change due to factors other than the FD distribution is the intensity change due to the electron DOS. Therefore, in step 2, the intensity change due to the electron DOS is excluded from the photoelectron spectrum measured in step 1. In other words, the intensity change due to the DOS of the photoelectron intensity is excluded by dividing the photoelectron spectrum by the electron DOS. Here, the electron DOS can be determined, for example, by the method described below. Figure 4 is a diagram showing a part of the band curve (Ek curve) that shows the electron band structure in the Γ-K direction of an Au(110) single crystal obtained by first-principles electronic structure calculation (band calculation). Figure 5 is a diagram obtained by counting the number of electron states every 10 meV in the energy range from -500 meV below the Fermi level to 500 meV above the Fermi level from the band curve in Figure 4. The obtained DOS is expressed as a function by fitting it with an appropriate function. In this way, the DOS for the electron energy of an Au(110) single crystal can be determined. 【0051】 The normalization of the photoelectron spectrum after eliminating the intensity change due to electron DOS is specifically performed by the function I(E:T,E) shown in equation (3) below. F This can be done using A, B). 【0052】 【number】 【0053】 In equation (3) above, I represents the intensity of the photoelectron spectrum, A represents the amplitude, and B represents the base. Typically, f is the FD distribution function. 【0054】 Figure 6 illustrates the procedure for eliminating intensity changes due to electron DOS and normalization using equation (3) above. Figure 6(a) corresponds to Figure 2. That is, Figure 6(a) shows the photoelectron spectrum of the Au(110) single crystal obtained in step 1. First, using equation (3) above, I is fitted to Figure 6(a) to obtain a provisional Fermi level E FDetermine this and divide it by the electron's DOS. This yields Figure 6(b). Again, by fitting I to Figure 6(b) using the above equation (3), the Fermi level E is obtained. F Simultaneously determining the photoelectron intensity of the photoelectron spectrum, the photoelectron intensity is normalized. Figure 6(c) shows the normalized photoelectron spectrum obtained in this way. Figure 6(c) corresponds to Figure 3. 【0055】 (Step 3) Next, in the normalized photoelectron spectrum, with the Fermi level as the reference zero, a certain electron energy E C For this, a pair of energy ranges (-E) are formed on the positive and negative sides in the axial direction of the electron energy E. C <E<E C Let R be defined as the positive range of R (0 <E<E C ) of R U The area in S U Defined as the negative range of R (-E C <E<0)のR L The area in S L Defined as, S L S for U Ratio S U / S L area ratio r ex This is defined as follows. In step 3, E C By changing the width of R, multiple area ratios r ex This is calculated. 【0056】 Figure 7 shows a certain electron energy E C When set, the energy range R and the positive range R U , negative range R L , area S U , and area S L This is a diagram to explain the following. Figure 7 shows the R etc. for the normalized photoelectron spectrum of Figure 3. As shown in Figure 7, the energy range R is defined as a certain electron energy E CIt is a range formed in a pair on the positive side and the negative side in the axial direction of the electron energy E with the Fermi level when set as the reference zero point. Therefore, the energy range R corresponds to the positive range R U on the positive side and the negative range R L on the negative side in the axial direction of the electron energy. That is, the width of R = the width of R U + the width of R L , and the width of R U = the width of R L is satisfied. The area S U is the area occupied by the photoelectron spectrum in the positive range R U in FIG. 7. The area S L is the area occupied by the photoelectron spectrum in the negative range R L in FIG. 7. 【0057】 In this embodiment, step 3 includes determining the upper limit value of the change in E C . By differentiating the normalized photoelectron spectrum with respect to energy and fitting a Gaussian function, the standard deviation σ can be calculated, and the upper limit value of the change in E C can be determined. 【0058】 FIG. 8 is a diagram for explaining a method of determining the upper limit value of the change in E C . FIG. 8 is obtained by differentiating the normalized photoelectron spectrum in FIG. 3 with respect to energy. By fitting a Gaussian function to FIG. 8, the standard deviation σ can be obtained, and the upper limit value of the change in E C can be determined. The upper limit value may be, for example, σ, 3σ, or 5σ. 【0059】 FIG. 9 is a graph plotting a plurality of area ratios r C calculated while changing E ex . FIG. 9 shows the relationship between the area ratio r ex and E C . 【0060】 (Step 4) In step 4, the electron energy E CA plurality of area ratios r calculated by varying ex For the function FDG obtained by performing a convolution integral of a Gaussian function with the FD distribution function represented by the following formula (1), the area ratio function r calculated using the following formula (2) th (E C ) By fitting, the thermodynamic temperature T and the broadening width ΔE of the sample surface are simultaneously determined. 【0061】 【Number】 【0062】 Here, the above formula (1) is a function FDG(E; T, ΔE) obtained by performing a convolution integral of a Gaussian function having a spectral broadening width, that is, a broadening width ΔE, as a broadening function with respect to the FD distribution having the thermodynamic temperature T as a variable. In the above formula (1), E is the electron energy, E F is the Fermi level, k B is the Boltzmann constant, and ε represents the mediating variable of the convolution integral. 【0063】 【Number】 【0064】 Here, the area ratio function r of the above formula (2) th (E C ) is a variable E representing the ratio of the area of the spectrum in the range above the Fermi level (0 < E < E C < E < 0) to the area of the spectrum in the range below the Fermi level (-E C ) with respect to the function obtained by performing a convolution integral of a Gaussian function with the FD distribution function represented by the above formula (1). C is a function of. 【0065】 Figure 10 is a graph showing as an example the area ratio function r th (E C ) for T = 50K, 100K, 300K assuming ΔE = 30 meV. Figure 10 shows the area ratio r th and E CThis shows the relationship. In step 4, the area ratio r calculated from the normalized photoelectron spectrum in Figure 9 is shown. ex and E C For the relationship between , the area ratio function r th (E C By fitting the sample, the thermodynamic temperature T and broadening width ΔE of the Au(110) single crystal surface can be determined simultaneously. 【0066】 (Step 5) The method according to this embodiment may include determining the thermodynamic temperature T using the least squares method (step 5). 【0067】 In step 5, χ is defined by the following equation (4). 2 The temperature accuracy dT of the thermodynamic temperature T may be determined using this method. 【0068】 【number】 【0069】 Here, in equation (4) above, n represents the number of data points and m represents the number of parameters. T^ is the optimal value of T determined by fitting, S(T) represents the least squares sum as a function of T, with the three fitting parameters other than T, calculated by the least squares method, fixed to their optimal values. S0(T^) represents the least squares sum with all four parameters fixed to their optimal values. 【0070】 Deviation Δχ 2 This is expressed as shown in equation (5) below and is calculated as a function of T. 【0071】 【number】 【0072】 Figure 11 shows the deviation Δχ. 2 This figure shows the relationship between the probability distribution of the deviation ΔT from the optimal value of the fitting parameter T. 2Table 1 shows the relationship between the value and the confidence level. 【0073】 [Table 1] 【0074】 As shown in Table 1, in step 5, the deviation Δχ 2 By setting = 1, it is possible to determine the temperature accuracy dT of the thermodynamic temperature T with a confidence level of 68.3%. 【0075】 In the method according to the above embodiment, the photoelectron spectrum of the sample surface can be obtained, for example, using an electron energy analyzer described in Patent Document 1. 【0076】 When the method according to the above embodiment is applied to a surface analyzer, material evaluation device, surface reaction control device, etc., simultaneous multi-functional measurement including the thermodynamic temperature of the sample surface becomes possible. These devices are widely used in technological fields such as advanced materials development, electronic device development, and chemical analysis, including the semiconductor industry. Therefore, the application of this method will contribute to the high functionality and improved reliability of these devices. Furthermore, the method according to the above embodiment is an absolute thermometer that does not require scale calibration by a temperature fixed point, and since the measurement area is limited to the surface at the atomic layer level of the material, it can be used as a new standard temperature measurement method (surface temperature measurement method) in the field of metrology standards. 【0077】 The embodiments described above are provided to facilitate understanding of the present invention and are not intended to limit it. Therefore, each element disclosed in the above embodiments is intended to include all design modifications and equivalents that fall within the technical scope of the present invention. For example, while the above embodiments describe the case where the sample is an Au(110) single crystal, the sample is not limited to an Au(110) single crystal. For example, the sample may be a metal or semiconductor other than an Au(110) single crystal. [Examples] 【0078】 The present invention will be described in more detail below with reference to examples, but the present invention is not limited in any way by the following examples. 【0079】 [Example 1] In Example 1, the photoelectron spectrum of the Au(110) single crystal surface was measured under liquid nitrogen cooling temperature using the method according to the above embodiment, and the thermodynamic temperature T and broadening width ΔE were determined. 【0080】 Figure 12A shows the E2000 photoelectron spectrum for Example 1. C Multiple area ratios r calculated by changing the value ex For this, the area ratio function r th (E C This figure shows the results of fitting the model. Through fitting, the optimal thermodynamic temperature T and broadening width ΔE for the Au(110) single crystal surface of Example 1 were determined to be T = 98.97 K and ΔE = 19.20 meV. The results are shown in Table 2. 【0081】 Next, the determined thermodynamic temperature T was evaluated using the above equation (5). Figure 12B shows the deviation Δχ for Example 1. 2 This figure shows the relationship between the and the fitting parameter T. From Figure 12B, for Example 1, the deviation Δχ 2 The temperature accuracy dT of the thermodynamic temperature T, determined with a confidence level of 68.3% by setting = 1, was ±0.60 K. The results are shown in Table 2. 【0082】 Table 2 shows the sensor temperature T of the Au(110) single crystal surface, measured under liquid nitrogen cooling conditions using a temperature sensor (LakeShore silicon diode sensor DT-670-SD). R This will also be shown. 【0083】 [Example 2] In Example 2, the photoelectron spectrum of the Au(110) single crystal surface was measured under liquid helium cooling temperature using the method according to the above embodiment, and the thermodynamic temperature T and broadening width ΔE were determined. 【0084】 Figure 13A shows the E2 photoelectron spectrum for Example 2. C Multiple area ratios r calculated by changing the value ex For this, the area ratio function r th (E C This figure shows the results of fitting the model. Through fitting, the optimal thermodynamic temperature T and broadening width ΔE for the Au(110) single crystal surface of Example 2 were determined to be T = 16.77 K and ΔE = 14.48 meV. The results are shown in Table 2. 【0085】 Next, the determined thermodynamic temperature T was evaluated using the above equation (5). Figure 13B shows the deviation Δχ for Example 2. 2 This figure shows the relationship between and the fitting parameter T. From Figure 13B, for Example 2, the deviation Δχ 2 The temperature accuracy dT of the thermodynamic temperature T, determined with a confidence level of 68.3% by setting = 1, was ±0.34 K. The results are shown in Table 2. 【0086】 Table 2 shows the sensor temperature T of the Au(110) single crystal surface, measured under liquid helium cooling conditions using a temperature sensor (LakeShore silicon diode sensor DT-670-SD). R This will also be shown. 【0087】 [Example 3] In Example 3, the photoelectron spectrum of the Au(110) single crystal surface was measured at room temperature using the method according to the above embodiment, and the thermodynamic temperature T and broadening width ΔE were determined. 【0088】 Figure 14A shows the photoelectron spectrum of Example 3, with E C Multiple area ratios r calculated by changing the value ex For this, the area ratio function r th (E C This figure shows the results of fitting the model. Through fitting, the optimal thermodynamic temperature T and broadening width ΔE for the Au(110) single crystal surface of Example 3 were determined to be T = 296.70 K and ΔE = 32.22 meV. The results are shown in Table 2. 【0089】 Next, the determined thermodynamic temperature T was evaluated using the above equation (5). Figure 14B shows the deviation Δχ for Example 3. 2 This figure shows the relationship between and the fitting parameter T. From Figure 14B, for Example 3, the deviation Δχ 2 The temperature accuracy dT of the thermodynamic temperature T, determined with a confidence level of 68.3% by setting = 1, was ±0.60 K. The results are shown in Table 2. 【0090】 Table 2 shows the sensor temperature T of the Au(110) single crystal surface, measured at room temperature using a temperature sensor (LakeShore silicon diode sensor DT-670-SD). R This will also be shown. 【0091】 [Table 2] 【0092】 [Comparative Example 1] In Comparative Example 1, the photoelectron spectrum of the Au(110) single crystal surface was measured under liquid nitrogen cooling temperature, and the FDG function of equation (1) above was fitted to the directly normalized photoelectron spectrum to obtain the thermodynamic temperature T C The thermodynamic temperature T of the Au(110) single crystal surface of Comparative Example 1 was determined. C is, T C The value was determined to be 90K. The results are shown in Table 3. 【0093】 Next, the thermodynamic temperature T was determined using the same method as in Example 1. C The following evaluation was performed. For Comparative Example 1, the deviation Δχ 2 The thermodynamic temperature T was determined such that the confidence level was 68.3% when set to =1. C Temperature accuracy dT C The value was ±3.2K. The results are shown in Table 3. 【0094】 Table 3 shows the sensor temperature T on the surface of an Au(110) single crystal. R This will also be shown. 【0095】 [Comparative Example 2] In Comparative Example 2, the photoelectron spectrum of the Au(110) single crystal surface was measured under liquid helium cooling temperature, and the FDG function of equation (1) above was fitted to the directly normalized photoelectron spectrum to obtain the thermodynamic temperature T C The thermodynamic temperature T of the Au(110) single crystal surface of Comparative Example 2 was determined. C is, T C The value was determined to be 35K. The results are shown in Table 3. 【0096】 Next, the thermodynamic temperature T was determined using the same method as in Example 2. C The following evaluation was performed. For Comparative Example 2, the deviation Δχ 2 The thermodynamic temperature T was determined such that the confidence level was 68.3% when set to =1. C Temperature accuracy dT C The value was ±1.9K. The results are shown in Table 3. 【0097】 Table 3 shows the sensor temperature T on the surface of an Au(110) single crystal. R This will also be shown. 【0098】 [Comparative Example 3] In Comparative Example 3, the photoelectron spectrum of the Au(110) single crystal surface was measured at room temperature, and the FDG function of equation (1) above was fitted to the directly normalized photoelectron spectrum to obtain the thermodynamic temperature T C The thermodynamic temperature T of the Au(110) single crystal surface of Comparative Example 3 was determined. C is, T C The value was determined to be 296K. The results are shown in Table 3. 【0099】 Next, the thermodynamic temperature T was determined using the same method as in Example 3. C The following evaluation was performed. For Comparative Example 3, the deviation Δχ 2 The thermodynamic temperature T was determined such that the confidence level was 68.3% when set to =1. C Temperature accuracy dT C The value was ±3.2K. The results are shown in Table 3. 【0100】 Table 3 shows the sensor temperature T on the surface of an Au(110) single crystal. R This will also be shown. 【0101】 [Table 3] 【0102】 As can be seen from the comparison between Table 2 and Table 3, the temperature accuracy of the example was significantly improved compared to the comparative example. Furthermore, the example yielded values ​​closer to the sensor temperature than the comparative example. Thus, according to the present invention, it is possible to provide a method for determining the thermodynamic temperature of a sample surface based on a photoelectron spectrum, which has improved temperature accuracy and is suitable for practical temperature measurement. [Industrial applicability] 【0103】 The method for determining the thermodynamic temperature of a sample surface based on the photoelectron spectrum of the present invention can determine the thermodynamic temperature of the sample surface with high accuracy, and therefore has great industrial potential.

Claims

[Claim 1] A method for determining the thermodynamic temperature of a sample surface based on photoelectron spectroscopy, Measuring the photoelectron spectrum of the sample surface, The measured photoelectron spectrum is used to eliminate intensity changes caused by factors other than the Fermi-Dirac (FD) distribution, and is further normalized. In the normalized photoelectron spectrum, with the Fermi level as the reference zero point, the energy E on the positive side of the axis direction of the electron energy E Ｃ and the negative energy -E Ｃ A range (-E Ｃ < E < E Ｃ ), defined as R, and among R, the range on the positive side (0 < E < E Ｃ ), the area of the photoelectron spectrum in R Ｕ is defined as S Ｕ and among R, the range on the negative side (-E Ｃ < E < 0), the area of the photoelectron spectrum in R Ｌ is defined as S Ｌ and when the ratio S Ｌ of S Ｕ is defined as the area ratio r Ｕ / S Ｌ while changing the energy E ｅｘ and calculating a plurality of area ratios r Ｃ while changing the width of R ｅｘ ; Multiple area ratios r calculated ｅｘ For this, the area ratio function r is calculated using equation (2) below from the function FDG obtained by convolving and integrating the FD distribution function with a Gaussian function, which is represented by equation (1) below. ｔｈ (E Ｃ By fitting the sample, the thermodynamic temperature T and broadening width ΔE of the sample surface can be determined simultaneously. A method that includes this. [Math 1] Here, the function FDG in equation (1) above is a function obtained by convolving and integrating an FD distribution function with thermodynamic temperature T as a variable with a Gaussian function with broadening width ΔE as the broadening function. In equation (1) above, E is the electron energy, E Ｆ is the Fermi level, k Ｂ ε represents the Boltzmann constant, and ε represents the parameter in the convolution integral. [Math 2] Here, the area ratio function r in equation (2) above ｔｈ (E Ｃ ) is calculated using the function that gives the theoretical spectral intensity obtained by equation (1) above, below the Fermi level (-E Ｃ For the area of ​​the spectrum in the range <E < 0, the Fermi level is (0 < E < E Ｃ The variable E represents the ratio of the area of ​​the spectrum within the range of ). Ｃ It is a function of . [Claim 2] The method according to claim 1, wherein the intensity change caused by factors other than the FD distribution is an intensity change due to the density of states (DOS) of electrons. [Claim 3] The method according to claim 1, wherein the photoelectron spectrum is normalized using the following formula (3). [Math 3] Here, in equation (3) above, I represents the intensity of the photoelectron spectrum, A represents the amplitude, and B represents the base. f may be the FD distribution function. [Claim 4] Multiple area ratios r ｅｘ To calculate the energy E, the standard deviation σ obtained by differentiating the normalized photoelectron spectrum with respect to energy and fitting it to a Gaussian function is used. Ｃ Determine the upper limit and the area ratio r ｅｘ The method according to claim 1, which includes calculating [a certain value]. [Claim 5] The method according to claim 1, further comprising determining the thermodynamic temperature T using the least squares method. [Claim 6] χ is defined by the following equation (4) ２ Using this, the deviation Δχ is calculated using the following formula (5). ２ The method according to claim 5, wherein the temperature accuracy dT of the thermodynamic temperature T is determined based on the above. [Math 4] Here, in equation (4) above, n represents the number of data points and m represents the number of parameters. T^ is the optimal value of T determined by fitting, and S(T) represents the sum of the least squares as a function of T, with the three fitting parameters other than T, calculated by the least squares method, fixed to their optimal values. ０ (T^) represents the least squares sum when all four parameters are fixed at their optimal values. [Math 5] Here, Δχ in equation (5) above ２ χ is calculated as a function of T. ２ This represents the deviation. [Claim 7] The method according to claim 1, wherein the sample is kept in a temperature range below the melting point of the sample.

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