Bosonic spectrum of a correlated multiband system, BaFe_1.80Co_0.20As₂, obtained via infrared spectroscopy

Figure 1 shows the measured $ab$ plane reflectance of BaFe_1.80Co_0.20As₂ at various temperatures in a temperature range of 8 - 300 K. The overall temperature-dependent trend of the reflectance of our over-Co-doped Ba122 is similar to the reported ones of Co-doped Ba122 systems tu:2010 ; kim:2010 ; lobo:2010 ; heumen:2010 , which were underdoped or optimally doped. However, the overall level of reflectance of our over-Co-doped Ba122 is higher than the reported ones. As we expected, the reflectance shows a metallic behavior, i.e., the reflectance is enhanced at low frequencies below $\sim$ 800 cm^-1 when the temperature decreases. At 8 K, the reflectance exhibited a sharp increase below $\sim$ 80 cm^-1, which is a signature for the formation of the SC gap (see Fig. S3 in the Supplementary Materials for a better view of the feature).

Fig. 2 shows the optical conductivity obtained from the measured reflectance using the Kramers-Kronig analysis. As shown in the figure, the temperature-dependent trends of typical and correlated metals in the low frequency region are not monotonic below 200 cm^-1 because the thermally excited phonons increase the scattering rate and the total spectral weight is conserved. This results in the temperature-dependent spectral weight redistribution. For the SC state, the superfluid spectral weight appears at $\omega=$ 0 because the scattering rate of the electrons (or spectral weight) involved in the superfluid is absolutely zero. The spectral weight does not appear in the finite frequency region and is called the ”missing spectral weight”, from which the superfluid spectral weight (or plasma frequency) can be estimated. The optical conductivities of the SC state (at 8 K and 15 K) show spectral weight suppressions below the SC gap, i.e., $\sim$ 50 cm^-1. In the inset in Fig. 2, a magnified view of the conductivity below 1000 cm^-1 is shown. In addition, a well-known infrared active phonon akrap:2009 can be seen near 260 cm^-1.

The superfluid plasma frequency ($\omega_{sp}$) of 8 K was obtained by two independent methods: one using the real part of optical conductivity ($\sigma_{1}(\omega)$), known as the Ferrell-Glover-Tinkham (FGT) sum rule glover:1956 ; ferrell:1958 ; tinkham:1975 , and the other using the imaginary part of optical conductivity ($\sigma_{2}(\omega)$) lee:2022a . In the FGT sum rule, the superfluid plasma frequency is described as $\omega_{sp}=\sqrt{(120/\pi)\int_{0^{+}}^{\infty}[\sigma_{1,N}(\omega)-\sigma_{1,SC}(\omega)]d\omega}$, where $\sigma_{1,N}(\omega)$ and $\sigma_{1,SC}(\omega)$ denote the real parts of optical conductivity of the normal and SC states, respectively. As shown in Fig. 3, the integration is equal to the hatched area in Fig. 3. In the second method, $\omega_{sp}=\lim_{\omega\rightarrow 0}\sqrt{[-\omega^{2}\varepsilon_{1}(\omega)]}$, where $\varepsilon_{1}(\omega)$ denotes the dielectric function, which is related to $\sigma_{2}(\omega)$ as $\varepsilon_{1}(\omega)=(60/\omega)\sigma_{2}(\omega)$. In the inset of Fig. 3, $-\omega^{2}\varepsilon_{1}(\omega)$ is displayed as a function of frequency. Note that the unit for all frequencies is cm^-1. By using the first and second methods, the superfluid plasma frequencies obtained are 7470 cm^-1 and 7710 cm^-1, respectively. The resulting $\omega_{sp}$ obtained by the two methods agree well with each other, even though the second method gives a slightly larger value than the first one. The London penetration depth ($\lambda_{L}$) can be obtained from the superfluid plasma frequency using the relation $\lambda=1/(2\pi\omega_{sp})$. The London penetration depth obtained using the FGT sum rule was 213.2 nm, while that obtained using the other method was 206.5 nm; these values are comparable to the previous results of Ba122 compounds li:2008 ; tu:2010 ; lobo:2010 ; heumen:2010 ; kim:2010 ; yoon:2017 .

The bosonic spectra were obtained from the measured optical conductivities at 27 K (normal state) using the extended Drude-Lorentz model lee:2022 and 8 K (SC state) using the two-parallel-channel approach hwang:2016 . In the extended Drude-Lorentz model, the complex optical conductivity ($\tilde{\sigma}(\omega)$) is described as follows lee:2022 :

$$\tilde{\sigma}(\omega)=\tilde{\sigma}_{\mathrm{ED}}(\omega)+\tilde{\sigma}_{\mathrm{L_{i}}}(\omega),$$

(1)

where $\tilde{\sigma}_{\mathrm{ED}}(\omega)$ denotes the extended Drude (ED) mode, which is obtained using the reverse process hwang:2015a , and $\tilde{\sigma}_{\mathrm{L_{i}}}(\omega)$ denotes the ith component of the Lorentz modes, which can be used to describe the interband transitions or phonon modes wooten ; tanner:2019 . In the two-parallel-channel approach, the complex optical conductivity ($\tilde{\sigma}(\omega)$) is described as follows hwang:2016 :

$$\tilde{\sigma}(\omega)=\tilde{\sigma}_{\mathrm{Ch1}}(\omega)+\tilde{\sigma}_{\mathrm{Ch2}}(\omega)+\tilde{\sigma}_{L_{i}}(\omega),$$

(2)

where $\tilde{\sigma}_{\mathrm{Ch1}}(\omega)$ and $\tilde{\sigma}_{\mathrm{Ch2}}(\omega)$ denote the complex optical conductivities of the channel 1 and channel 2, respectively. Note that each channel is a SC channel. $\tilde{\sigma}_{\mathrm{Ch1}}(\omega)$ and $\tilde{\sigma}_{\mathrm{Ch2}}(\omega)$ were obtained the reverse process hwang:2015a , in which the optical scattering rate, $1/\tau^{op}(\omega)$, (or the imaginary part of the optical self-energy, $-2\Sigma^{op}_{2}(\omega)$) were obtained from an input bosonic spectrum using the generalized Allen’s formulas, the real part of the optical self-energy ($-2\Sigma^{op}_{1}(\omega)$) was obtained from the imaginary one using the Kramers-Kronig relation between the real and imaginary parts of the optical self-energy, and eventually, the complex optical conductivity was obtained using the ED model formalism.

The imaginary part of the optical self-energy for the ED mode ($-2\Sigma^{op}_{\mathrm{ED},2}(\omega)$) is written as followslee:2022 :

	$\displaystyle-2\Sigma^{op}_{\mathrm{ED},2}(\omega)$	$\displaystyle=$	$\displaystyle\frac{\pi}{\omega}\int_{0}^{\infty}d\Omega\>I^{2}B(\Omega)\Big{[}2\omega\coth\Big{(}\frac{\Omega}{2T}\Big{)}-(\omega+\Omega)\coth\Big{(}\frac{\omega+\Omega}{2T}\Big{)}$		(3)
		$\displaystyle+$	$\displaystyle(\omega-\Omega)\coth\Big{(}\frac{\omega-\Omega}{2T}\Big{)}+\frac{1}{\tau^{op}_{\mathrm{imp}}}\Big{]},$

where $I^{2}B(\Omega)$ denotes the bosonic spectrum. Here $I$ denotes the coupling constant between the electron and a force mediated boson. $1/\tau^{op}_{\mathrm{imp}}$ denotes the impurity scattering rate. The imaginary part of the optical self-energy for each SC channel ($-2\Sigma^{op}_{\mathrm{Ch_{i}},2}(\omega)$) is written as followshwang:2016 :

$$-2\Sigma^{op}_{\mathrm{Ch_{i}},2}(\omega)=\int^{\infty}_{0}\!\!d\Omega\>I^{2}B(\Omega)\>K(\omega,\Omega)+\frac{1}{\tau^{op}_{\mathrm{imp}}(\omega)},$$

(4)

where $i$ denotes the channel number, which is either 1 or 2. $K(\omega,\Omega)$ denotes the kernel, which is written as follows allen:1971 :

	$\displaystyle K(\omega,\Omega)$	$\displaystyle=$	$\displaystyle\frac{2\pi}{\omega}(\omega-\Omega)\Theta(\omega-2\Delta_{\mathrm{Ch_{i}},0}-\Omega)$		(5)
		$\displaystyle\times$	$\displaystyle\!\!\!E\Big{(}\frac{\sqrt{(\omega-\Omega)^{2}\!-\!(2\Delta_{\mathrm{Ch_{i}},0})^{2}}}{\omega\!-\!\Omega}\Big{)},$

where $\Theta(\omega)$ represents the Heaviside step function (i.e., 1 for $\omega\geq 0$ and 0 for $\omega<0$), and $E(x)$ represents the complete elliptic integral of the second kind, where $x$ is dimensionless. $\Delta_{\mathrm{Ch_{i}},0}$ denotes the SC gap for the channel $i$. $1/\tau^{op}_{\mathrm{imp}}(\omega)$ denotes the impurity scattering rate for the $s$-wave SC state, which is frequency-dependent and is written as followsallen:1971 :

$\displaystyle\frac{1}{\tau^{op}_{imp}(\omega)}$

$\displaystyle=$

$\displaystyle\!\!\frac{1}{\tau^{op}_{\mathrm{imp}}}E\Big{(}\frac{\sqrt{\omega^{2}\!-\!(2\Delta_{\mathrm{Ch_{i}},0})^{2}}}{\omega}\Big{)}.$

(6)

Note that the same $I^{2}B(\omega)$ was used for the two channels as in the previous studies hwang:2016 ; lee:2022a .

Once the imaginary part of the optical self-energy is obtained, the corresponding real part can be obtained using the Kramers-Kronig relation between them, expressed as follows hwang:2016 :

$$-2\Sigma^{op}_{1}(\omega)=-\frac{2}{\pi}P\int^{\infty}_{0}\frac{\Omega[-2\Sigma^{op}_{2}(\Omega)],}{\Omega^{2}-\omega^{2}}d\Omega$$

(7)

where $P$ denotes for the principal part of the improper integral. Here, $-2\Sigma^{op}_{2}(\Omega)$ can be either $-2\Sigma^{op}_{\mathrm{ED},2}(\omega)$ or $-2\Sigma^{op}_{\mathrm{Ch_{i}},2}(\omega)$, depending upon the electronic (either normal or SC) state. Eventually, the complex optical conductivity ($\tilde{\sigma}(\omega)\equiv\sigma_{1}(\omega)+i\sigma_{2}(\omega)$) was obtained for the extended Dude mode and the two SC channels using the ED model formalism as follows gotze:1972 ; allen:1977 ; puchkov:1996 ; hwang:2004 :

$$\tilde{\sigma}(\omega)=i\frac{\Omega_{p}^{2}}{4\pi}\frac{1}{\omega+[-2\tilde{\Sigma}^{op}(\omega)]},$$

(8)

where $\Omega_{p}$ denotes the plasma frequency of charge carriers, which can be different for two different channels, and $\tilde{\Sigma}^{op}(\omega)$ $(\equiv\Sigma^{op}_{1}(\omega)+i\Sigma^{op}_{2}(\omega))$ denotes the complex optical self-energy. Here, $-2\tilde{\Sigma}^{op}(\Omega)$ can be either $-2\tilde{\Sigma}^{op}_{\mathrm{ED}}(\omega)$ or $-2\tilde{\Sigma}^{op}_{\mathrm{Ch_{i}}}(\omega)$, depending on the electronic (either normal or SC) state.

The model $I^{2}B(\omega)$ was used in this study; it consists of two Gaussian peaks: one is sharp and located at a low frequency, and the other is broad and located at a high frequency. The same model bosonic spectrum has been used in previous studies as well lee:2022 ; lee:2022a . Here, the sharp Gaussian is an optical mode, which may be associated with the magnetic resonance mode observed by the inelastic neutron scattering inosov:2010 . Fig. 4 shows data and fit at 27 K (normal state) using the extended Drude-Lorentz model described above in a wide spectral range up to 8000 cm^-1. The resulting bosonic spectrum ($I^{2}B(\omega)$) is shown in Fig. 6 (see the red dashed line). An extended Drude mode, two Lorentz modes, and an interband transition were needed for the fitting. Note that the interband transition consists of three Loremtz modes. The plasma frequency ($\Omega_{\mathrm{ED},p}$) and impurity scattering rate ($1/\tau^{op}_{\mathrm{imp}}$) for the ED mode were found to be 1.23 eV and 15 meV, respectively. The the inset of Fig. 4 displays a magnified view of the ED mode and the two Lorentz modes in the blue dashed line located at low frequencies below 1000 cm^-1. Interestingly, two out of the five Lorentz modes are located in a very low frequency region below $\sim$400 cm^-1; these two modes were named low-energy Lorentz modes, as shown in the inset. Nevertheless, these low-energy Lorentz modes are absent in the K-doped Ba122 systems dai:2013 ; lee:2022 ; lee:2022a (see also Fig. S4 and S5 in the Supplementary Materials) but have been observed in a similar frequency region by previous optical studies of Co-doped Ba122 systems lobo:2010 ; heumen:2010 . The origin of low-energy Lorentz modes is not yet clarified. Lobo et al. speculated that these Lorentz modes are the response of localized carriers induced by disorders lobo:2010 . The disorder is related to the FeAs plane lattice distortion caused by Co-doping in the Co-doped Ba122 systems because these low-energy Lorentz modes are absent in the K-doped Ba122 systems dai:2013 ; lee:2022 ; lee:2022a , where the FeAs plane is intact by K-doping.

Fig. 5 shows data and fit at 8 K (SC state) using the two-parallel-channel approach described above in a wide spectral range up to 8000 cm^-1. The resulting bosonic spectrum ($I^{2}B(\omega)$) is shown in Fig. 6 (see the blue solid line). Two SC channels (Ch1 and Ch2), the low-energy Lorentz modes, and the interband transition were considered for the fitting. The SC gaps ($\Delta_{Ch1,0}$ and $\Delta_{Ch2,0}$) and the plasma frequencies ($\Omega_{Ch1,p}$ and $\Omega_{Ch2,p}$) for the two SC channels were observed to be 3.6 and 8.0 meV, and 806 and 558 meV, respectively. The same impurity scattering rate ($1/\tau^{op}_{\mathrm{imp}}$) of 15 meV was used for both channels in Eq. (6). The bosonic spectra of the two channels were assumed to be the same as in previous literature hwang:2016 . The inset in Fig. 5 displays a magnified view of the optical conductivities for the two SC channels and the low-energy Lorentz modes located in the low frequency region below 1000 cm^-1. The low-energy Lorentz modes at the two temperatures (8 and 27 K) were observed to be almost identical. Note that the data could not be properly fitted without including the low-energy Lorentz modes.

Fig. 6 shows the resulting $I^{2}B(\omega)$ at 8 K (SC state) and 27 K (normal state). Note that the resulting $I^{2}B(\omega)$ shows a similar temperature-dependent trend and a similar spectral width as in previous literature wu:2010 . Various physical quantities, such as the coupling constant ($\lambda$), the maximum SC transition temperature ($T^{Max}_{c}$), the SC coherence length ($\xi_{SC}$) and the upper critical field ($H_{c2}$), can be obtained from $I^{2}B(\omega)$. The coupling constant was obtained from its definition, i.e., $\lambda\equiv 2\int_{0}^{\omega_{c}}I^{2}B(\Omega)/\Omega\>d\Omega$, where $\omega_{c}$ is a cutoff frequency, 100 meV in our case. The obtained coupling constants were 2.589 and 1.383 at 8 K and 27 K, respectively. $T^{Max}_{c}$ at 8 K was obtained using the generalized McMillan formula hwang:2008c , i.e., $k_{B}T_{c}\cong 1.13\>\>\hbar\omega_{\mathrm{ln}}\>\exp[-(1+\lambda)/(g\lambda)]$, where $k_{B}$ denotes the Boltzmann constant, $T_{c}$ denotes the SC transition temperature, $\hbar$ denotes the reduced Planck’s constant, $g$ denotes an adjustable parameter between 0 and 1.0, and $\omega_{\mathrm{ln}}$ denotes the logarithmic averaged frequency of $I^{2}B(\omega)$, which is defined by $\omega_{\mathrm{ln}}\equiv\exp[(2/\lambda)\int_{0}^{\omega_{c}}\mathrm{ln}\Omega\>I^{2}B(\Omega)/\Omega\>d\Omega$]. If $g=$ 1.0, $T_{c}$ will be its maximum value, $T^{Max}_{c}$. At 8 K, $\omega_{\mathrm{ln}}$ was found to be 10.41 meV, and $T^{Max}_{c}$ was found to be 34.1 K, which is larger than the measured value of 22.0 K. Note that the $T_{c}^{Max}$ is a possible maximum $T_{c}$, not a real maximum $T_{c}$. The observation indicates that the $I^{2}B(\omega)$ is sufficient for explaining superconductivity in the Co-doped Ba122 sample. The time scale of the retarded interaction between electrons via exchanging the mediated bosons is contained in $I^{2}B(\omega)$ hwang:2021 . The time scale can be obtained from the first moment of $I^{2}B(\omega)/\omega$, i.e., $\langle\Omega\rangle\equiv(2/\lambda)\int_{0}^{\omega_{c}}\omega^{\prime}[I^{2}B(\omega^{\prime})/\omega^{\prime}]d\omega^{\prime}$. The SC coherence length, $\xi_{SC}$, can be obtained from the time scale and the Fermi velocity ($v_{F}$). It can be written in terms of $\langle\Omega\rangle$ and $v_{F}$ as $\xi_{SC}=(1/2)v_{F}[2\pi/\langle\Omega\rangle](1/2\pi)=(1/2)[v_{F}/\langle\Omega\rangle]$ (or $=(1/2)[\hbar v_{F}/\hbar\langle\Omega\rangle]$) hwang:2021 . The estimated $\hbar\langle\Omega\rangle$ was 12.05 meV at 8 K. The reported average Fermi velocity of Co-doped Ba122 systems richard:2010 ; brouet:2012 , i.e., $\hbar v_{F}\cong$ 0.5 eVÅ, was used. The estimated SC coherence length was 20.75 Å, which is comparable to the reported value of Co-doped Ba122 kano:2009 . The upper critical field ($H_{c2}$) can also be estimated from the SC coherence length ($\xi_{SC}$) using the Ginzburg–Landau expression, i.e., $H_{c2}=\Phi_{0}/(2\pi\xi_{SC}^{2})$ kittel:2005 , where $\Phi_{0}$ denotes the flux quantum. The estimated upper critical field was found to be $\sim$ 75.6 T, which is consistent with the previous results of Co-doped Ba122 hanisch:2015 . The Ginzburg-Landau parameter ($\kappa$) is defined by the two characteristic length scales ($\lambda_{L}$ and $\xi_{SC}$) for superconductivity as $\kappa\equiv\lambda_{L}/\xi_{SC}$ cyrot:1973 . The estimated $\kappa$ was 99.5, indicating that the Co-doped sample is a type-II superconductor.

The results of the electron-doped BaFe_1.80Co_0.20As₂ were compared with those of a hole-doped Ba_0.49K_0.51Fe₂As₂. Both samples were similarly overdoped. The SC transition temperatures were 22.0 K and 34.0 K for the electron- and hole-doped Ba122 samples, respectively. The results of the hole-doped sample have been reported previously lee:2022 ; lee:2022a . The energy-dependent distribution of the bosonic spectra of the two systems was observed to be different. The spectral weight of the bosonic spectrum of the Co-doped Ba122 was located in a lower energy region compared with that of the K-doped one (see Fig. 6 and Fig. S6 in the Supplementary Materials). This resulted in a larger coupling constant ($\lambda$) of the electron-doped Ba122 (2.59) than that of the hole-doped one (1.68) at 8 K lee:2022a . Both the average time scale ($2\pi/\langle\Omega\rangle=$ $2\pi$/(12.05 meV)) and Fermi velocity ($\hbar v_{F}=$ 0.5 eVÅ) of the bosonic spectrum of the Co-doped Ba122 are larger than those ($2\pi$/(18.72 meV) and 0.38 eVÅ) of the K-doped one, resulting in a longer SC coherence length ($\xi_{SC}=$ 20.75 Å) for the Co-doped Ba122 compared with that (10.15 Å) of the K-doped one lee:2022a . The $\xi_{SC}$ reflects the size of the Cooper pair, which was larger in the electron-doped Ba122 than that in the hole-doped one. The $\lambda_{L}$ (206.5 nm) of the Co-doped Ba122 was slightly smaller than that (221.7 nm) of the K-doped one lee:2022a . Note that the London penetration depths were obtained using the superfluid plasma frequencies estimated using $\lim_{\omega\rightarrow 0}[-\omega^{2}\varepsilon_{1}(\omega)]$. The electron-doped Ba122 exhibits roughly half of the $\kappa$ of the hole-doped one because the former has a twice larger $\xi_{SC}$ but a similar $\lambda_{L}$ to the hole-doped one. Therefore, the electron-doped Ba122 is a weaker type-II superconductor compared with the hole-doped one. Moreover, the electron-doped Ba122 has the low-energy Lorentz modes, whereas the hole-doped one does not have any such modes (see Figs. 4 and 5 and Figs. S2 and S3). This difference is due to the doping methods because electron-doping occurs by replacing the Fe atom in the FeAs plane with a Co atom of a different size from the Fe atom, i.e., the FeAs plane is disordered by Co-doping. However, the FeAs plane, which is known as the charge transport layer, remains intact through hole-doping because the Ba atom in the Ba layer outside the FeAs plane is replaced with a K atom, owing to which K-doped Ba122 is one of the cleanest Fe-based superconductors. Therefore, a finite impurity scattering rate (15 meV) was necessary for fitting the data of electron-doped Ba122, whereas no impurity scattering rate was necessary for fitting the data of hole-doped Ba122 lee:2022 ; lee:2022a . An additional difference between electron- and hole-doped Ba122 is the Lindhard function caused by the different-type carriers, resulting in different effective masses for different Fermi surface sheets neupane:2011 . We speculate that the differences between the two Ba122 systems describe above may result in different $I^{2}B(\omega)$. Consequently, the different $I^{2}B(\omega)$ gives rise to the different SC $T_{c}$ between the two Ba122 systems.