Transport properties of the metallic state of overdoped cuprate superconductors from an anisotropic marginal Fermi liquid model

We consider the implications of a phenomenological model self-energy for the charge transport properties of the metallic phase of the overdoped cuprate superconductors. The self-energy is the sum of two terms with characteristic dependencies on temperature, frequency, location on the Fermi surface, and doping. The first term is isotropic over the Fermi surface, independent of doping, and has the frequency and temperature dependence characteristic of a Fermi liquid. The second term is anisotropic over the Fermi surface (vanishing at the same points as the superconducting energy gap), strongly varies with doping (scaling roughly with $T_c$, the superconducting transition temperature), and has the frequency and temperature dependence characteristic of a marginal Fermi liquid. Previously it has been shown this self-energy can describe a range of experimental data including angle-dependent magnetoresistance and quasiparticle renormalizations determined from specific heat, quantum oscillations, and angle-resolved photoemission spectroscopy. Without introducing new parameters and neglecting vertex corrections we show that this model self-energy can give a quantitative description of the temperature and doping dependence of a range of reported transport properties of Tl${}_2$Ba${}_2$CuO${}_{6+\delta }$ samples. These include the intralayer resistivity, the frequency-dependent optical conductivity, the intralayer magnetoresistance, and the Hall coefficient. The temperature dependence of the latter two are particularly sensitive to the anisotropy of the scattering rate and to the shape of the Fermi surface. In contrast, the temperature dependence of the Hall angle is dominated by the Fermi liquid contribution to the self-energy that determines the scattering rate in the nodal regions of the Fermi surface.

Figure 1

Frequency dependence of the imaginary part of the model self-energy. The first part ${\Sigma }_{\mathrm{FL}}$, is Fermi-liquid-like, with a quadratic $\omega$ and $T$ dependence, up to the high-$\omega$ cutoff ${\omega }_{\mathrm{FL}}^{*}$, and with a prefactor $s/{\omega }_{\mathrm{FL}}^{*2}$, This part is taken to be doping independent. The second part is the anisotropic marginal Fermi liquid part, whose imaginary part is constant for $\omega <\pi T$ and proportional to $T$, while it is linear in $\omega$ for higher $\omega$ up to a high-$\omega$ cutoff ${\omega }_{\mathrm{AMFL}}^{*}$. Its strength is given by $\lambda \left(\phi \right)$, which is strongly anisotropic over the Fermi surface and also strongly doping dependent.

Figure 2

Comparison of the measured temperature dependence of the intralayer resistivity ${\rho }_{xx}$ to fits using the self-energy model. Fits are performed in the $T$ range from above $T_c$ to 200 K and agree nicely with the data. At higher $T>200$ K our model predicts a stronger increase in ${\rho }_{xx}$, which could be improved by smoother high-frequency cutoffs in the self-energy. This would introduce new parameters and is beyond the scope of this paper. $T_c$ values and references for the data denoted with A and B are given in Table .

Figure 3

Doping ($p$) or transition temperature ($T_c$) dependence of the model parameters extracted from fitting the temperature dependence of the resistivity for a range of Tl2201 samples. The fitted model parameters are consistent with the values and doping dependencies extracted from ADMR.[26] Strength of the impurity scattering $1/\left(2{\tau }_0\right)$ shows additional decreasing trend with decreasing doping, which might be due to a smaller amount of doped interstitial oxygen. The AMFL strength $\lambda$ shows a strong increase with decreasing doping in good agreement with the results from ADMR.[26] Good agreement with the self-energy model is also found for the doping-independent strength of the FL-like scattering $s/{\omega }_{\mathrm{FL}}^{*2}$. For the doping dependence of $T_c$ we use the phenomenological relation (see Sec. 2). $T_c$ values and references for the data denoted with A, B, C, and D are given in Table .

Figure 4

Comparison of the measured optical absorption spectra $A\left(\omega \right)$ for Tl2201 at three different dopings with the model self-energy. The latter is for parameter values extracted from ADMR.[26] A good description of the $\omega$, $T$, and $T_c$ dependencies of $A\left(\omega \right)$ is obtained for $\omega \lesssim 1000$ cm${}^{-1}$. For higher $\omega$, $A\left(\omega \right)$ shows a stronger increase with $\omega$, which could be improved with a softer high-$\omega$ cutoff for the self-energy. This is similar to what is found for the high-$T$ dc resistivity in Fig. 2. Data for $T_c=$ 15 K and 70 K are from Ref. 24 and data for $T_c=23$ K are from Ref. 23.

Figure 5

(Left) Three different approximations for the shape of the Fermi surface (FS). The FS deduced from ADMR measurements,[12] ARPES measurements,[8] and tight-binding approximation to the LDA calculations[26, 37] are denoted with ADMR, ARPES, and LDA, respectively. (Right) Curves of the mean-free path $\mathbf{l}\left(\phi \right)$ as one goes around the FS for the three different FSs. According to Ong's geometric interpretation[47] the encircled area is proportional to ${\sigma }_{xy}^{\left(1\right)}$ [Eq. (29)]. Although the shapes of the FSs do not change much between different approximations, the mean-free paths and the encircled areas in $\mathbf{l}$ space change substantially. The main difference comes from the curvature of the FS $({\partial }_{\phi }{\mathbf{v}}_{0,F}\left(\phi \right))$ close to $\phi \sim \pi /8$. The absolute value of $R_H$ is therefore very sensitive to the shape of the FS. The mean-free path $\mathbf{l}\left(\phi \right)$ was calculated with our model self-energy for $T=100$ K, and $T_c=30$ K.

Figure 6

Calculated temperature dependence of the Hall coefficient ($R_H$) for three slightly different Fermi surfaces (see Fig. 5). The absolute change or maximal value of $R_H$ depends strongly on the shape of the Fermi surface. All curves are calculated with the same scattering rate, using our self-energy model for $T_c=30$ K. Temperature broadening effects due to the Fermi-Dirac distribution are not taken into account since they are small ($<10$$\%$). The slightly different values of $R_H$ at $T=0$, where the scattering is dominated by impurities and is therefore isotropic, also comes from small changes in the Fermi surface shape.

Figure 7

Temperature dependence of the measured Hall coefficient $R_H$ for several different $T_c$s. These all correspond to dopings for which there is a large hole Fermi surface and show a nonmonotonic temperature dependence that increases with increasing $T_c$. For $T\lesssim T_c$ $R_H$ may be strongly suppressed by the superconducting transition. Data for $T_c=$ 0 K and 81 K are for polycrystalline samples measured in Ref. 21, data for $T_c=$ 10 K and 50 K are from Ref. 17, $T_c=$ 15 K data are from Ref. 18, and $T_c=$ 25 K data are from Ref. 20.

Figure 8

Temperature dependence of the Hall coefficient $R_H$ calculated with our model self-energy for several $T_c$s and for the ADMR Fermi surface.[12] Results should be compared with Fig. 7. The figure illustrates how decreasing doping (increasing $T_c$ or $\lambda$ or anisotropy of the self-energy) leads to a large change in the magnitude of the temperature dependence. All results are obtained for the same Fermi surface and only the anisotropy of the self-energy is changed. Arrows shown in the lower right indicate the weak dependence on the band structure change with doping, indicating the absolute shift of the zero temperature value of $R_H$ ($T_c=0$) for two different dopings ($p=0.15$ and $p=0.30$) as given by a rigid band shift.

Figure 9

Temperature dependence of the Hall angle $\cot {\theta }_H$. (Top) Experimental data show that $\cot {\theta }_H$ has a linear dependence on $T^2$ and a weak doping dependence. A similar dependence is found for our model (bottom), where the linear dependence of $\cot {\theta }_H$ on $T^2$ comes predominantly from the Fermi-liquid-like scattering (${\Sigma }_{\mathrm{FL}}^{\prime \prime }$) that gives the scattering rate on the nodal part of the Fermi surface (which also gives the dominant contribution to the Hall conductivity; see Fig. 5). (Top) The inset shows a small down-turned deviation from the $T^2$ dependence of $\cot {\theta }_H$ at low $T$ (for $T_c=15$ K). This is also obtained within our model (bottom inset), although not as pronounced as in the experimental data. Experimental data for $T_c=$ 10 K and 50 K are from Ref. 17. $T_c=$ 15 K data is from Ref. 18 and $T_c=$ 25 K data is from Ref. 20. Our model results are calculated with the ADMR Fermi surface.

Figure 10

Temperature dependence of the intralayer magnetoresistance $\Delta {\rho }_{xx}^{\left(2\right)}/{\rho }_{xx}$ for various $T_c$ (or strength of anisotropic scattering) calculated with the AMFL model for the ADMR Fermi surface and for $B_z=10$ T. Temperature dependence of the result for $T_c=0$ resembles the $T$ dependence of the isotropic scattering, while the anisotropy induces the variation from this result with similar $T$ dependence as observed in $R_H$ (see Fig. 8). The magnetoresistance strongly depends on the Fermi surface shape (see inset and Fig. 5). All results are calculated for the fixed chemical potential.

Figure 11

Comparison between the measured intralayer magnetoresistance $\Delta {\rho }_{xx}^{\left(2\right)}/{\rho }_{xx}$ at $B_z=10$ T for two samples[20] and the result of the AMFL model for $T_c=25$ K. AMFL results are calculated for the ADMR and LDA Fermi surfaces and agree qualitatively with the measured data.

Figure 12

Plot to test modified Kohler's rule. Temperature dependence of $(\Delta {\rho }_{xx}^{\left(2\right)}/{\rho }_{xx}){\cot }^2{\theta }_H$, calculated for the AMFL model, is shown for various $T_c$. Results show weak $T$ dependence, which is in agreement with the experimental observations shown with the dash-dotted black line (ratio $\sim 1.9$).[16] Results were calculated with the ADMR Fermi surface and depend strongly on the shape of the Fermi surface. In the inset we show results for the LDA Fermi surface, which show weaker variation with $T$ and a quantitative agreement with experimental data.

Figure 13

Small temperature dependence of the factor ${(1+\frac{1}{1+b/a})}^{-1}$ appearing in $\cot {\theta }_H$. This factor shows less than 10$\%$ variation with $T$ from 50 to 300 K. Therefore the main $T$ dependence of $\cot {\theta }_H$ comes from the isotropic scattering, which in our model self-energy is $\propto T^2$ (in agreement with experiment).

Figure 14

Comparison of the imaginary part of the self-energy at zero frequency $(-{\Sigma }^{\prime \prime }(\omega =0))$ from CDMFT results (Ref. 67, Fig. 11) with our model self-energy. CDMFT results were obtained for different doping levels $p=0.3,0.25,$ and $0.2$ and for different patches on the Fermi surface (“nodal” denotes nodal patch, while “antin.” denotes antinodal patch). CDMFT results are only available at higher temperatures ($T>200$ K) due to limitations of the quantum Monte Carlo method used. Our model self-energy is most reliable at low $T$. [Experiments suggest it should become more linear for $T>200$ K (see Fig. 2)]. Most reliable comparison with CDMFT can therefore be done at $T=200$ K. At such $T$ CDMFT predicts a weaker isotropic self-energy (compare CDMFT $n=0.70$ and our nodal self-energy). Quantitative comparison for stronger anisotropies or lower doping is more difficult due to patch averaging in CDMFT. However, CDMFT predicts the correct trend with doping and order of the magnitude for the self-energy. Our antinodal self-energy was calculated with $T_c=60$ K, and the energy scale of CDMFT data was set with the hopping parameter $t_1=0.438$ eV.

Figure 15

Comparison of results for the Matsubara frequency dependence of $\mathrm{Im}\Sigma \left(i{\omega }_n\right)$ from CDMFT (Ref. 67, Fig. 8) with our model self-energy. It is seen that the saturated value of the self-energy is similar in both results. The isotopic CDMFT result ($p=0.28$) predicts a smaller slope at low frequencies (and thus are a quasiparticle weight closer to one) than our model. Parameters for our model self-energy are the same as in Fig. 14 and the temperature corresponds to that of the CDMFT calculation, $T=0.05t_1\sim 250\mathrm{K}$.

Figure 16

Inverse square root of the intralayer magnetoresistance, ${(\Delta {\rho }_{xx}^{\left(2\right)}/{\rho }_{x}x)}^{-1/2}$ vs $T^2$. With this choice of the axes the $T^2$ behavior of ${(\Delta {\rho }_{xx}^{\left(2\right)}/{\rho }_{x}x)}^{-1/2}$ becomes more apparent (at least at high $T$) and shows similar behavior to $\cot {\theta }_H$. Therefore the ratio of the two is expected to show weak $T$ dependence and obeys the modified Kohler's rule as already discussed in the main text and shown in Fig. 12. Curves are calculated with the ADMR Fermi surface and for several $T_c$s.