Thesis High-Resolution Photoemission Study of Kondo Insulators ...

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20 Chapter 2. Photoemission Spectroscopy The Fourier component of the temperature Green function is expressed as Gr(iωl) = � α ′ α ′′ e β(Ω−E′ ) eβ(E′ −E ′′ )+1 iωl + E ′ − E ′′ < α ′′ |a † r |α′ > (2.13) where Ω is the thermodynamic potential. By substituting a general complex number z for a discrete iωl and exchanging ′ and ′′ , we have and ImGr(z) = � Gr(z) = � α ′ α ′′ e eβ(Ω−E′ ) [β(E ′ −E ′′ )+1] z − (E ′ − E ′′ ) < α ′′ |ar|α ′ >. (2.14) α ′ α ′′ e β(Ω−E′ ) β(E (e ′ −E ′′ ) ′ † +1)< α ′′ |ar|α ′ >δ(z − (E ′ − E ′′ )). (2.15) Therefore, the imaginary part of the temperature Green function multiplied by the Fermi-Dirac (FD) function leads to the photoemission spectrum at finite temperature, which is given by averaging the initial eigenstates with the Boltzmann factors [2.1, 4]: P < (ω) ∝ 1 � π α ′ α ′′ e β(Ω−E′ ) ′ † < α ′′ |ar|α ′ >δ(ω − (E ′ − E ′′ )) = 1 ( eβω 1 )( +1 π )ImGr(ω) = f(ω)A(ω). (2.16) where f(ω) = 1 eβω is the FD function. Similarly, the inverse-photoemission spectrum +1 at finite temperature is given by P > (ω) ∝ (1 − f(ω))A(ω). 2.1.2 Derivation of Spectral DOS from Observed Photoemission Spectra In Eqs. (2.16) we have an interest in the temperature dependence of the spectral DOS ImGr(ω) rather than that of FD function and we can obtain a spectral DOS of the whole energy range by dividing either photoemission spectra or inverse-photoemission spectra by each distribution factor. For simplicity, we deal with the photoemission spectra hereafter3 . Experimentally observed spectra are broadened with a finite energy resolution: � Pobs(ω) ∝ f(ω1)A(ω1)R(ω − ω1)dω1 (2.17) 3 We note that experimentally the energy resolution is over ten times worse for the inversephotoemission spectroscopy than for photoemission spectroscopy.
2.1. General Principles 21 where R(ω) is a Gaussian function4 . In the case for a constant spectral DOS of A(ω) = A, Eq. (2.17) reduces to � Pobs(ω) ∝ A f(ω1)R(ω − ω1)dω1 (2.18) and division of the observed photoemission spectra by the FD function convoluted with the instrumental resolution gives the spectral DOS [2.6]. DOS Intensity DOS DOS broadened 6K 75K 305K 6 K 75 K 305 K -80 -40 0 40 Energy relative to E F (meV) Figure 2.2: Top: Calculated B p partial DOS of YbB12 (solid curve) [2.7] with the same DOS convoluted with the 7 meV Gaussian (dashed curve). Middle: Photoemission spectra for the DOS in the top panel at 6 K, 75 K, and 305 K. Bottom: Spectral DOS deduced from the spectra in the middle panel. To show the validity of such a division for a spectral DOS with structures, we have applied that process to the calculated DOS of YbB12 [2.7]. The top panel of Fig. 2.2 4 Fourier transformation for the convolution of two functions yields the product of the two functions. Shin et al. [2.5] eliminated the effect of the instrumental resolution by such transformation.
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Page 4 and 5: ii CONTENTS 5.2 Experimental . . .
Page 6 and 7: 2 Chapter 1. Introduction Figure 1.
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Page 10 and 11: 6 Chapter 1. Introduction peaks, wh
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Page 16 and 17: 12 Chapter 1. Introduction activati
Page 18 and 19: 14 References [1.14] F. Iga, M. Kas
Page 21 and 22: Chapter 2 Photoemission Spectroscop
Page 23: 2.1. General Principles 19 Photoele
Page 27 and 28: 2.2. Experimental 23 DOS Intensity
Page 29 and 30: 2.2. Experimental 25 Figure 2.5: Il
Page 31 and 32: 2.2. Experimental 27 baking out the
Page 33: References [2.1] O. Gunnarsson and
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Page 48 and 49: 44 References the Kondo peak positi
Page 50 and 51: 46 Chapter 4. Evolution of Electron
Page 63 and 64: References [4.1] G. Aeppli and Z. F
Page 65: References 61 [4.25] P. Weibel,M. G
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70 Chapter 5. Temperature Dependenc
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72 Chapter 5. Temperature Dependenc
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74 References [5.10] N. Shimizu, F.
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Chapter 6 Substitution and Temperat
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6.1. Introduction 79 Figure 6.2: Ma
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6.3. Results and Discussion 81 6.3
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6.3. Results and Discussion 83 DOS
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6.3. Results and Discussion 85 dist
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6.3. Results and Discussion 87 Apar
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6.3. Results and Discussion 89 of E
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6.4. Summary 93 and consistent anal
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96 References [6.11] M. J. Rozenber
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98 Chapter 7. Temperature and Co-Su
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100 Chapter 7. Temperature and Co-S
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References [7.1] G. Aeppli and Z. F
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Chapter 8 Temperature and Al-Substi
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8.2. Experimental 113 and Fisk [8.2
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8.3. Results and Discussion 115 8.3
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8.3. Results and Discussion 119 dep
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8.3. Results and Discussion 121 σ
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8.4. Summary 123 temperature magnet
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126 References [8.14] T. Tonogai, A
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128 Chapter 9. Conclusion has shown
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130 Chapter 9. Conclusion (a) FeSi
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132 Chapter 9. Conclusion developed
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