Jahn-Teller Effect and Spin-Orbit Coupling: Friends or Foes?

The Jahn-Teller effect is one of the most fundamental phenomena important not only for physics but also for chemistry and material science. Solving the Jahn-Teller problem and taking into account strong electron correlations we show that quantum entanglement of the spin and orbital degrees of freedom via spin-orbit coupling strongly affects this effect. Depending on the number of $d$ electrons, it may quench (electronic configurations $t_{2g}^2$, $t_{2g}^4$, and $t_{2g}^5$), partially suppress ($t_{2g}^1$), or, in contrast, induce ($t_{2g}^3$) Jahn-Teller distortions. Moreover, in certain situations, interplay between the Jahn-Teller effect and spin-orbit coupling promotes formation of the “Mexican hat” energy surface facilitating various quantum phenomena.

Popular Summary

Aristotle once wrote that nature abhors a vacuum, a statement that was modified in the early 20th century by physicists Jahn and Teller, who showed that nature also does not like to have too much symmetry. They explained that materials often avoid crystalline structures that are too symmetric and that couplings between electrons and the lattice sometimes drive crystals away from highly symmetric configurations. We show that this Jahn-Teller effect can be strongly modified when one takes into account the orbital motion of the electrons. The resulting spin-orbit coupling can, depending on the number of electrons, suppress or induce Jahn-Teller distortions.

Mathematically, this effect is very easy to understand. Each interaction aims to put electrons on their own orbits, or wave functions. The spin-orbit coupling chooses those wave functions with maximal orbital moment (spherical harmonics), while the Jahn-Teller effect tends to minimize the overall electrostatic repulsion (to occupy cubic harmonics). As in any human society, these two competing interests need to find a compromise.

The spin-orbit coupling is especially strong for heavy metals of the fifth and sixth row of the periodic table (partially filled $4d$ and $5d$ shells), such as molybdenum, tungsten, or iridium, and our results will help us to understand various structural distortions in materials based on such ions. Moreover, we show that the combination of Jahn-Teller physics and strong spin-orbit coupling can lead to very specific quantum effects, which can be potentially used for quantum computations.

Figure 1

Diagrams of crystal-field splitting due to the Jahn-Teller effect without spin-orbit coupling for different number of electrons: (a) $d^1$, (b) $d^2$, (c) $d^3$, (d) $d^4$, and (e) $d^5$.

Figure 2

Results for the $t_{2g}^1$ configuration ($g=B=1$). (a)–(c) Evolution of the energy surface dependence on $Q_2$ and $Q_3$ modes as a function of the spin-orbit coupling constant $\lambda$. (d) Amplitude of distortion $u$ as a function of $\lambda$ at the $(0,\pm u)$ points in the $Q_2{Q}_3$ plane. (e) Energies of compressed and elongated octahedra as function of $\lambda$.

Figure 3

Level splitting diagrams due to spin-orbit coupling for different number of electrons, the case of the $jj$ coupling scheme (very large $\lambda$).

Figure 4

The amplitude of the Jahn-Teller distortions for $t_{2g}^3$ configuration as a function of the ratio of the spin-orbit coupling strength $\lambda$ and Hund’s exchange $J_H$, the results of the exact diagonalization calculations for $g=B=1$ and $J_H=0.5$.

Figure 5

Configuration $d^3$. Energy surfaces for different ratios of $\lambda$ and $J_H$, $g=B=1$. One may note three JT minima for $\lambda =J_H$ (corresponding to compressed octahedra), but already for $\lambda =3J_H$ they become nearly indistinguishable and the energy surface resembles the Mexican hat.

Figure 6

The amplitude of the Jahn-Teller distortions for $t_{2g}^5$ configuration as a function of spin-orbit coupling strength $\lambda$. Results of the exact diagonalization calculations for $g=B=1$. Insets demonstrate how the total energy surface as a function of the $Q_2$ and $Q_3$ distortions evolves with $\lambda$.

Figure 7

(a) Four solutions corresponding to the total energy minimum of a pair of Jahn-Teller centers in a common corner geometry ($t_{2g}^1$ configuration, $\lambda =0$). $Q_3$, $Q_2^b$, and $Q_2^t$ are given in units of $g/B$. (b) Distortions (and orbitals) corresponding to one of these minima.

Figure 8

The amplitude of the distortion for a pair of Jahn-Teller centers as a function of $\lambda$ ($g=B=1$). There are four equivalent distortions of two octahedra for $\lambda =0$, see Fig. 7. Top and bottom octahedra are distorted differently in each case (due to competition between local JT and coupling between them). Increase of the SOC strength reduces this difference.