Common Fermi-liquid origin of $T^2$ resistivity and superconductivity in $n$-type SrTiO${}_3$

A detailed analysis is given of the $T^2$ term in the resistivity observed in electron-doped SrTiO${}_3$. Band-structure data are presented that provide values for the bare mass, density of states, and plasma frequency of the quasiparticles as functions of doping. It is shown that these values are renormalized by approximately a factor of two due to electron-phonon interaction. It is argued that the quasiparticles are in the antiadiabatic limit with respect to electron-phonon interaction. The condition of antiadiabatic coupling renders the interaction mediated through phonons effectively nonretarded. We apply Fermi-liquid theory developed in the 70’s for the $T^2$ term in the resistivity of common metals, and combine this with expressions for $T_c$ and with the Brinkman-Platzman-Rice (BPR) sum rule to obtain Landau parameters of $n$-type SrTiO${}_3$. These parameters are comparable to those of liquid ${}^3$He, indicating interesting parallels between these Fermi liquids despite the differences between the composite fermions from which they are formed.

Figure 1

Temperature dependence of the resistivity, the inverse Hall constant, and the mobility of SrTi${}_{1-x}$Nb${}_x$O${}_3$ for different carrier concentrations.

Figure 2

Temperature-dependent relaxation rates of SrTi${}_{1-x}$Nb${}_x$O${}_3$ for four different carrier concentrations, using the relation $\rho \left(T\right)=4\pi {\omega }_p^{-2}{\tau }^{-1}$. Plasma frequencies, ${\omega }_p$, are obtained from the Drude spectral weight measured with infrared spectroscopy[34] and listed in Table . Inset: difference between experimental data and fitted curve, $\chi \left(T\right)=\hslash /\tau \left(T\right)-\hslash /{\tau }_{\text{fit}}\left(T\right)$, demonstrating upward departure from $T^2$ behavior of the resistivity above 100 K.

Figure 3

Band dispersion of the lowest unoccupied bands of SrTiO${}_3$ in the low-temperature tetragonal phase. The directions in momentum space are labeled according to the high-temperature cubic Brillouin zone, so that $[1,0,0]$ corresponds to momentum along the Ti-O bond direction. The rightmost panel indicates the position of the Fermi energy as a function of carrier concentration. $x_{c1}=4.0\times {10}^{-5}$ and $x_{c2}=2.6\times {10}^{-3}$ are critical carrier concentrations where the Fermi energy enters the second and the third bands.

Figure 4

Fermi surface of the high-temperature cubic phase at 2% doping, showing the large anisotropy of the lowest band. At the critical doping $x_c=0.097$, a topological transition takes place where the Fermi surfaces open up along the three axes.

Figure 5

Enlarged view of Fig. 3 indicating the position of the Fermi level for $x=3.6\times {10}^{-4}$ charge carriers. The cor-responding value $ka=0.134$ is in excellent agreement with the hitherto unexplained de Haas-van Alphen frequency reported by Gregory et al.[41]

Figure 6

Top panel: doping dependence of the density of states (DOS) at the Fermi energy. The solid curve corresponds to the tight-binding band structure fitted to the wien2k ab initio band structure, with parameters of the first row of Table . Squares,[42] circles,[33, 43] pentagon:[44] density of states obtained from the linear term in the specific heat. Middle panel: doping dependence of the Drude spectral weight, ${\omega }_p^2$. The solid curve corresponds to the band-structure results. Diamonds are the experimental values.[34] Bottom panel: (i) ratio of experimental DOS over band structure DOS (experimental and theoretical values taken from top panel, the meaning of the symbols is the same). (ii) Ratio of band structure over experimental ${\omega }_p^2$ (values taken from middle panel). (iii) Ratio of bare and experimental (dressed) Fermi velocity, $v_{F,b}/v_{F,e}$ (triangle, data from Ref. 37). The grey curve is a smooth interpolation, $m^{*}/m_b=2.0+1.2\exp (-x/0.005)$.

Figure 7

The quantity $k_{F}l$, where $l$ is the mean free path, calculated using $k_{F}l=2{\epsilon }_F^{*}{\hslash }^{-1}\tau$, and using the experimental $\tau$ of Fig. 2 and the calculated Fermi energies corrected by the mass renormalization of Fig. 6. The high values of $k_{F}l$ below 100 K imply the itinerant character of the charge carriers.

Figure 8

Top panel: $T_c$ of a large number of samples. Pentagons: data by Binnig et al.[4] Circles: data by Koonce et al.[3] Middle panel: coherence volume times the density of Cooper pairs. The high value indicates that each pair is overlapping with a huge number of other pairs. Lower panel: doping dependence of the coupling constants ${\lambda }_{\tau }$ obtained from the $T^2$ term of the resistivity and ${\lambda }_0$ from $T_c$. Circles: ${\lambda }_{\tau }$ using $a_2$ listed in the second column of Table  and Eq. (3). Diamonds: idem using $a_2^{\prime }$. Bars: coupling constants $\lambda$ from $T_c$ at the corresponding carrier concentration of SrTi${}_{1-x}$Nb${}_x$O${}_3$ using the $T_c$’s of the top panel and Eq. (C5), ${\epsilon }_F^{*}={\epsilon }_F{m}_b/m^{*}$ using ${\epsilon }_F$ from Fig. 3 and $m^{*}/m_b$ from Fig. 6. Upper (lower) limits of the bars indicate the value for $g$ obtained with ${\omega }_c=5$ meV (80 meV).

Figure 9

Solutions for $A_1^s$, $A_0^a$, $A_1^a$, the triplet superconducting coupling constant, and the mass enhancement as a function of $A_0^s$ when taking the experimentally determined ${\lambda }_{\tau }=0.38$, ${\lambda }_0=0.15$, and the BPR sum rule as constraints assuming singlet pairing for the ground state.