Dimensional crossover in layered $f$-electron superlattices

Motivated by the remarkable experimental realizations of $f$-electron superlattices, e.g., CeIn${}_3$/LaIn${}_3$ and CeCoIn${}_5$/YbCoIn${}_5$ superlattices, we analyze the formation of heavy electrons in layered $f$-electron superlattices by means of the dynamical mean field theory. We show that the spectral function exhibits formation of heavy electrons in the entire system below a temperature scale $T_0$. However, in terms of transport, two different coherence temperatures $T_x$ and $T_z$ are identified in the in-plane and the out-of-plane resistivity, respectively. Remarkably, we find $T_z<T_x\sim T_0$ due to scatterings between different reduced Brillouin zones. The existence of these two distinct energy scales implies a crossover in the dimensionality of the heavy electrons between two and three dimensions as temperature or layer geometry is tuned. This dimensional crossover would be responsible for the characteristic behaviors in the magnetic and superconducting properties observed in the experiments.

Figure 1

The spectral function $A(\omega =0,\mathit{k})$ for $(L_A,L_B)=(2,5)$ at (a) $k_z=0$, (b) $k_z=\pi /2L$, (c) $k_y=0,$ and (d) $k_y=\pi /2$. Temperature is $T=0.0015$. Violet corresponds to high-intensity regions and black corresponds to low-intensity regions.

Figure 2

The Fermi surface with $U=0,$ $V=0$ for (a) $c$ electrons at $K_z=q_1^c+k_z=\pi /2=2\pi /7+3\pi /14$, (b) $f$ electrons, (c) $c$ electrons at $k_y=\pi /2$, and (d) $f$ electrons at $k_y=\pi /2$. The Fermi surfaces in (a) and (c) are the same. Violet corresponds to high-intensity regions and black corresponds to low-intensity regions.

Figure 3

The spectral function $A(\omega ,\mathit{k})$ at $(k_y,k_z)=(0,\pi /2L)$ for several temperatures when $(L_A,L_B)=(2,5)$. Temperatures for (a), (b), (c), and (d) are $T=0.02$, 0.015, 0.01, and 0.0015, respectively. Violet corresponds to high-intensity regions and black corresponds to low-intensity regions.

Figure 4

The spectral function $A(\omega ,\mathit{k})$ at $(k_x,k_y)=(0.55\pi ,0)$ for several temperatures when $(L_A,L_B)=(2,5)$. Temperatures for (a), (b), (c), and (d) are $T=0.02$, 0.015, 0.01, and 0.0015, respectively.

Figure 5

The spectral function $A(\omega ,\mathit{k})$ for $U=0$ (a) at $(k_y,k_z)=(0,\pi /2)$ and (b) at $(k_x,k_y)=(0.55\pi ,0)$ for $(L_A,L_B)=(2,5)$.

Figure 6

Temperature dependence of the resistivity for the $x$ direction, ${\rho }_{xx}$ (left panel), and the $z$ direction, ${\rho }_{zz}$ (right panel). The numbers $(L_A,L_B)$ in the figure denote corresponding layer configurations. For a comparison, the resistivities for the 3D PAM $(L_A,L_B)=(1,0)$ are also shown with red curves.

Figure 7

Temperature dependence of the $x$-axis conductivity (left panel) and $z$-axis conductivity (right panel) for $(L_A,L_B)=(2,5)$.

Figure 8

Coherence temperatures vs $L_A/L$ for $(L_A,L_B)=(1,0)$, (4,1), (3,1), (2,1), (2,2), (2,3), and (2,5).