Metallization of solid molecular hydrogen in two dimensions: Mott-Hubbard-type transition

We analyze the pressure-induced metal-insulator transition in a two-dimensional vertical stack of ${\text{H}}_2$ molecules in (x-y) plane, and show that it represents a striking example of the Mott-Hubbard-type transition. Our combined exact diagonalization approach, formulated and solved in the second quantization formalism, includes also simultaneous ab initio readjustment of the single-particle wave functions, contained in the model microscopic parameters. The system is studied as a function of applied side force (generalized pressure), both in the ${\text{H}}_2$-molecular and $\text{H}$-quasiatomic states. Extended Hubbard model is taken at the start, together with longer-range electron-electron interactions incorporated into the scheme. The stacked molecular plane transforms discontinuously into a (quasi)atomic state under the applied force via a two-step transition: the first between molecular insulating phases and the second from the molecular to the quasiatomic metallic phase. No quasiatomic insulating phase occurs. All the transitions are accompanied by abrupt changes of the bond length and the intermolecular distance (lattice parameter), as well as by discontinuous changes of the principal electronic properties, which are characteristic of the Mott-Hubbard transition here associated with the jumps of the predetermined equilibrium lattice parameter and the effective bond length. The phase transition can be interpreted in terms of the solid hydrogen metallization under pressure exerted by, e.g., the substrate covered with a monomolecular ${\text{H}}_2$ film of the vertically stacked molecules. Both the Mott and Hubbard criteria at the insulator to metal transition are discussed.

Figure 1

Schematic representation of stacked vertically ${\text{H}}_2$ molecular $2\text{D}$ layer forming square lattice. The bond length and the intermolecular distance are marked by $R$ and $a$, respectively. There are eight atoms in the supercell (dark blue spheres). The supercell is repeated periodically to conform periodic boundary conditions (PBC). Shaded spheres indicate atoms which are continuations resulting from the PBC implementation. The indices $\alpha$, $\beta$ distinguish the component atoms of each molecule.

Figure 2

Schematic representation of nonzero hoppings (a) and electron-electron interaction terms (b), taken into account in the Hamiltonian (14). The corresponding nonzero terms included are listed explicitly and represent the matrix elements between site pairs marked also as the solid or bold circles. Note that the parameters $t^{\alpha \beta }=K^{\alpha \beta }=0$ determine the range of the corresponding dynamical processes accounted for in an exact manner.

Figure 3

Ground-state energy per molecule as a function of the bond length (intramolecular distance) $R$ for four selected values of the lattice parameter $a$. The minima are marked by the vertical arrows.

Figure 4

Enthalpy (per molecule) versus pressure $p$. At lower pressure, two molecular phases are stable; the transition to the quasiatomic phase occurs at $p_{c2}\sim 0.1954\text{Ry}/a_0^2$, as marked. $E_B(p=0)=-2.3858\text{Ry}$, $R_{\mathrm{eff}}(p=0)=1.4031a_0$, and $a(p=0)=4.3371a_0$. Thin lines extrapolate the enthalpies of the particular phases beyond the regime of their stability. Insets show some detail on the transitions. For details, see main text.

Figure 5

Intermolecular distance (lattice parameter) $a$ for $2\text{D}$ bilayer crystal as a function of pressure $p$. The transitions are clearly of discontinuous (first-order) nature at temperature $T=0$. Note a spectacular decrease of lattice parameter by $8.47\%$ (corresponding to $16.22\%$ volume decrease) at the transition ($p_{c2}$) from molecular to quasiatomic phase. The thin lines denote the lattice parameter of the phases in the regime, where they are not of the lowest enthalpy. The arrows mark the jump of the intermolecular distance at the transitions with the increasing pressure.

Figure 6

Intramolecular distance (bond length) $R$ as a function of pressure $p$. An abrupt change by $70.69\%$ at the transition from molecular to quasiatomic state (at $p_{c2}$) is clearly visible. The spectacular increase of the optimized bond length $R=R_{\mathrm{eff}}$ at $p_{c2}$ is taking place towards quasiatomicity (cf. Fig. 5). Only a small difference between $R_{\mathrm{eff}}$ in both of the molecular phases ($3.21\%$—close to $p_{c1}$) is observed. The arrows mark the interatomic distance jump at $p_{c1}$ and $p_{c2}$ when increasing $p$.

Figure 7

Effective Bohr radius $1/\zeta$ of the renormalized Slater orbitals composing the Wannier functions for a $2\text{D}$ system, as a function of pressure $p$. The atomic function size changes by $27.02\%$ at the transition to the molecular phase II and by $29.97\%$ at the transition to the quasiatomic (metallic) state. The arrows mark the Slater-orbital size jumps when increasing $p$.

Figure 8

Estimate of the Hubbard gap, $U-W$, with the bare bandwidth $W$ computed for the single-electron part of Hamiltonian (14) as a function of $p$ in the molecular and quasiatomic correlated states. The bandwidth changes radically at $p_{c2}$. The negative gap value means that the two bands overlap and hence the system is in metallic state (for a detailed discussion, see main text). The arrows mark the sequence of jumps with the increasing pressure.

Figure 9

Ratio between the intra-atomic (Hubbard) repulsion amplitude $U$ and the lower-bandwidth $W$ in the correlated state, as a function of $p$. Both quantities are calculated for the renormalized orbitals composing the Wannier functions. At the critical pressures the ratio jumps: from the value 1.3880 to 1.5778 at $p_{c1}$ (at the transition between the two molecular phases), and from 1.3112 to 0.6000 at $p_{c2}$, i.e., at the transition to the quasiatomic phase. The latter defines the Mott-Hubbard-type transition to a moderately correlated state. Close to the transition, even in the molecular phases the value of the bare bandwidth $W$ is not decisively smaller than $U$. The arrows mark the jumps when increasing the pressure.

Figure 10

Principal hopping correlation functions $〈{\widehat{c}}_{i\mu }^{†}{\widehat{c}}_{j\nu }^{}〉$ versus pressure $p$. $〈{\widehat{c}}_{0\alpha }^{†}{\widehat{c}}_{0\beta }^{}〉$ corresponds to the intramolecular hopping and $〈{\widehat{c}}_{0\alpha }^{†}{\widehat{c}}_{1\alpha }^{}〉$ to the intermolecular one. Note that some of the straight lines coincide. Note that whereas for the molecular crystal the dominant hopping is $〈{\widehat{c}}_{0\alpha }^{†}{\widehat{c}}_{0\beta }^{}〉=1$ and the remaining one is almost equal to zero, for the quasiatomic phase the presented correlation functions in the metallic state are almost equal to those for free electrons, i.e., $\approx 1/2$. Such a behavior provides us with a clear sign of both quasiatomic nature and metallic character of the highest-pressure state, as the renormalized hoppings are practically the same and equal to $\frac{1}{2}$.

Figure 11

Electronic density $n\left(\mathbf{r}\right)$ in 3D near the molecular I (a) → molecular II (b) transition at $p_{c1}=0.1102\text{Ry}{a}_0^{-2}$. The ellipsoidal character of density is a signature of ${\text{H}}_2$ molecular states with the symmetric character (with respect to the molecule center of mass) of its spatial distribution.

Figure 12

Electronic density $n\left(\mathbf{r}\right)$ in 3D near the molecular II (a) → quasiatomic (b) transition ($p_{c2}=0.1954\text{Ry}{a}_0^{-2}$). Note a clear changeover from the molecular ellipsoidal (top) to the quasiatomic (spherical) configuration shape of the density, characteristic for symmetric-in-space molecular states and quasiatomic nature of the single-particle states, respectively. Also, the electronic-density profiles illustrate directly the character of the Mott-Hubbard transition at $p=p_{c2}$.

Figure 13

(a),(b) Dispersion relations for bare bands at the molecular I → molecular II transition ($p_{c1}=0.1102\text{Ry}/a_0^2$). (c),(d) The same at the molecular II → quasiatomic transition ($p_{c2}=0.1954\text{Ry}/a_0^2$). One expects that the lowest band in (a)–(c) will split additionally into the Hubbard subbands in those states as then $U/W>1.5$, whereas in the state (d) they will overlap as the $U/W\approx 0.5$, i.e., the system eventually becomes a moderately correlated metal.

Figure 14

Enthalpy isolines on the plane $a\text{−}R$ at the border between I and II states (a) and at the II-quasiatomic border (b). The points mark the minima with the arrow connecting them as a guide to the eye. Note that both transitions involve primarily a radical change in the effective molecule size $R_{\mathrm{eff}}$.

Figure 15

Single-electron wave functions $w_0^{\beta }(\mathbf{r}=(x,0,-R/2))$, in the planar direction (a) and $w_0^{\beta }(\mathbf{r}=(0,0,z-R/2))$ (b) (along the molecule, $z$ direction) in the molecular phases I $[a=2.76261\left(a_0\right)$, $R_{\mathrm{eff}}=1.1511\left(a_0\right)$, $\zeta =0.8571\left(a_0^{-1}\right)]$ and II $[a=2.67911\left(a_0\right)$, $R_{\mathrm{eff}}=1.1881\left(a_0\right)$, $\zeta =1.0564\left(a_0^{-1}\right)]$ near the transition (at $p_{c1}=0.1102\text{Ry}/a_0^2$).

Figure 16

Single-electron wave functions $w_0^{\beta }(\mathbf{r}=(x,0,-R/2))$ (a) and $w_0^{\beta }(\mathbf{r}=(0,0,z-R/2))$ (b) (along the $z$ direction) in molecular phase II $[a=2.43783\left(a_0\right)$, $R=1.1296\left(a_0\right)$, $\zeta =1.0887\left(a_0^{-1}\right)]$ and in the quasiatomic phase $[a=2.23133\left(a_0\right)$, $R=1.9281\left(a_0\right)$, $\zeta =0.7398\left(a_0^{-1}\right)]$ near the transition (at $p_{c2}=0.1954\text{Ry}/a_0^2$). Note the shrinking in the quasiatomic phase.