Liquid Semiconductor Proposal:

What governs the electronic structure of liquid semimetals?

Grant Goodyear

Department of Chemistry

University of Houston

Houston, TX 77204

With this research program my goal is seemingly quite simple: predict whether or not a molten semimetal (metalloid), such as liquid silicon or liquid germanium, is a metal or an insulator, and endeavor to understand why that should be the case. In the solid state this problem is a relatively simple one to solve (the picture at right is the crystalline Si band structure¹), despite an explicit need to treat Avogadro's number of atoms. In the liquid state, however, the situation is considerably more complicated, because the electronic structure depends on the liquid's structure, which in turn depends upon the electronic interactions between particles.² Liquid semimetals are particularly difficult to handle³ because locally their structure is not necessarily, even upon averaging, radially symmetric (i.e., not hard-sphere packing) ¾ being Group IV atoms, they prefer to form tetrahedral local structures (an ordering that is difficult to obtain without including at least three-body interactions). Exactly what effect does the strange combination of medium-range disorder and local orientational ordering have on the liquid’s electronic structure? This proposal is still somewhat in a nascent stage, but my current plans are as follows.

First Year ¾ Tight-binding band structure of liquid Si and Ge. To what extent does the electronic structure follow from simply getting the liquid structure right? Non-ideal hard spheres can generate the requisite tetrahedral packing. Doped semimetals. How do dopants change the liquid’s electronic structure?

Third Year ¾ Inclusion of electron correlation: Hubbard and Pariser-Parr-Pople models. Compare with ab-initio (Car-Parrinello⁴) results, which put in electron correlation explicitly.

Fifth Year ¾ Can we develop a local-environment-specific hybridization (valence-bond) picture that efficiently builds in the electronic-structure/liquid-structure interplay? Both simulation and traditional liquid theories would be necessary (going beyond a tight-binding description via simulation is expensive).

Funding sources: NSF and PRF.

Introduction

Certainly one of the most basic questions that one can ask about a substance is whether or not it conducts electricity. In the case of a crystal the answer is relatively straightforward to compute.¹ Tight-binding (molecular-orbital) predictions are at least qualitatively correct (it’s trivial to explain the conductance of Group I metals in terms of a “half-filled band”), and the fact that one needs to account for Avogodro’s number of atoms turns out to be essentially irrelevant due to the periodicity of the crystal. In fact, it is rather difficult to overstate the importance of spatial periodicity, which causes the reciprocal lattice vector k to be a good quantum number, in terms of making crystalline band structure calculations tractable (k provides symmetry information analogous to the SALC’s one would obtain from a molecular point group). Moreover, in a crystal the disparity between thermal energies (~ 25 meV) and electronic energies (1⁺ eV) is so large, and the atomic zero-temperature bands.

Liquids break all of these “rules”. Liquids are inherently disordered, so k is hardly a good quantum number (there is no useful symmetry). Although the thermal and electronic energy scales are still quite disparate, cooperative behavior in a liquid can nonetheless strongly influence the liquid’s electronic structure. In fact, one can see from the metal/non-metal transition in simple liquid metals (such as Hg, Cs, or Rb)² that fairly minor changes in thermodynamic conditions (usually by changing the density, experimentally) can dramatically alter a macroscopic property of the liquid. These differences from what happens in crystals makes liquid band structure considerably more difficult to handle. Of course, the result is that liquids are more versatile (and more fun) than crystals.

To put this problem in perspective, according to the Born-Oppenheimer approximation, the nuclear positions of the atoms in a fluid are frozen on the timescale of the electrons in that fluid. Consequently, computing the electronic structure of a liquid requires that for each equilibrium set of nuclear positions we compute the molecular orbitals (in a tight-binding model) of the entire system, and only then are we allowed to average over equilibrium nuclear positions. To be precise, the electronic tight-binding Hamiltonian for an equilibrium set of nuclear positions is given by

, (1)

where j, k = 1, …, N denote atoms, m, n = 1, …, n denote (orthonormal) atomic orbitals, are the atomic orbital energies, and are the hopping energies for an electron to hop from orbital m on atom j to orbital n on atom k. The electronic density of states is then given by

, (2)

with {E_m(R)} the set of one-electron energy levels produced by diagonalizing the Hamiltonian (for a single liquid configuration R), and the angle brackets denoting an equilibrium average over liquid configurations. Admittedly the problem is “just” that of diagonalizing a matrix, but here we are stuck with the fact that the matrix involved is of the order of Avogadro’s number, and we have to perform that diagonalization for each liquid configuration in the equilibrium average. Clearly this problem is not a simple one.

What makes liquid semimetals particularly interesting is that (1) they have obvious technological implications, since the Group IV semimetals underlie the current “computer revolution” and (2) they feature directional bonding that makes them quite different from simple (radially-symmetric) liquids. Indeed, liquid Si, while no longer exhibiting the tetrahedral coordination of the solid phase, nonetheless has a liquid coordination number of 6 that is still much lower than the closest-packed value of 12.^5,6 (Interestingly, Si also increases in density by about 10% upon melting.⁷) Moreover, liquid Si is a metal,^5,6,8 whereas crystalline (and amorphous) Si, of course, is a semiconductor. Relatively little is known about liquid Ge, although recent XANES⁹ and Car-Parrinello^9,10 studies suggest that the liquid is more metallic than the solid.

Car-Parrinello methods:

Car-Parrinello⁴ (or ab initio Molecular Dynamics) methods are the metaphorical equivalent of throwing everything, including the kitchen sink, at the problem. A molecular dynamics simulation can certainly provide the equilibrium set of nuclear positions required for Eq. (1), but the Car-Parrinello technique goes even further ¾ solving the full electronic problem (including electron correlation) at each time step of the simulation using Density Functional methods. Even better, the nuclear potential energies (and thus their forces) are determined by the ground-state solution of the electronic problem. The net result is that as long as the Born-Oppenheimer approximation holds (as well as the LDA approximation of density functional theory, and a large enough plane-wave basis set is used) then the Car-Parrinello methods return the exact quantum-mechanical electronic structure and classical dynamics of the liquid under study. Unsurprisingly, such state-of-the-art calculations are extremely computationally intensive (a measure of their difficulty is that one state point generally corresponds to one paper in the literature). Nonetheless, one of the earlier Car-Parrinello studies looked specifically at liquid Si,⁵ and recently liquid Ge studies have begun to appear.^9,10 As with any simulation method, the numerical accuracy of the method tends to be offset somewhat by the difficulty involved in attempting to sift through the mountains of information that the simulation invariably produces.

Computer Simulation + Tight-Binding

A simpler simulation-based method is to replace the density functional computation of Car-Parrinello with a simple tight-binding calculation. Empirical nuclear potential energies are usually employed, with the contribution from the instantaneous electronic structure either ignored altogether, or included using the Hellmann-Feynman theorem.^6,11 Molecular dynamics or Monte Carlo methods can then be used to propagate the nuclear degrees of freedom. Although not ab initio, such methods produce results significantly faster than do their ab initio counterparts. Tight-binding molecular dynamics results for liquid Si agree very well with the considerably more labor-intensive Car-Parrinello results for the same system,⁶ and this method provides an excellent check of the more approximate methods that I wish to employ.

Integral Equation Solutions

Although tight-binding computer simulations are still an order of magnitude faster than ab initio computer simulations, they are still relatively slow, requiring more time than an ordinary classical computer simulation, producing more data than one can easily wade through, and necessarily including a number of particles that is far smaller than Avogadro’s number. The latter point is particularly important in studying conductivity, since the finite size of the system makes it difficult to determine whether or not the measured electrical cooperativity is truly macroscopic ¾ that is, does the system really conduct over an “infinite” distance?

Instead, I plan to employ integral-equation methods to solve this problem (see my Supercritical Fluids proposal for a quick overview of liquid theories). Thanks to separate tours-de-force by Winn & Logan¹² and Xu & Stratt,¹³ it is possible to rigorously perform the diagonalizations of Eq. (1) in the process of solving the usual classical integral equations, except that the electronic degrees of freedom become coupled to the nuclear degrees of freedom in the integral equations, making the equations self-consistent. The net result is that instead of having to propagate 10²³ degrees of freedom in a computer simulation (diagonalizing at every time step), we need only solve a single integral equation. Although not quite as straightforward, it has also been possible to solve Hubbard-like models (meaning non-zero electron correlation) using these methods.¹⁴ To the best of my knowledge, the only integral-equation solution to address semi-metals is that of López-Martin, Lomba, Kahl, Winn, and Rassinger,¹¹ who compute the band structure for three state points (and a single model) of liquid Si; their integral equation results agreed very well with those of tight-binding molecular dynamics.

My plan for studying liquid semimetals via integral equations effectively falls into two parts. Initially, I intend to take the already existing tight-binding theories, and use them to “play” with the systems of interest. Integral equation are quick to solve, once the original formalism is developed, so we should be able to answer a lot of questions. How does the electronic structure of semimetals depend on the thermodynamic conditions? Can we adequately predict the electronic structure of molten Ge? To what extent does the electronic structure follow from simply getting the liquid structure right? The López-Martin, et al results¹¹ suggest that, at least for liquid Si, directional bonding plays a negligible role in the electronic structure; is the same true in liquid Ge? Using non-ideal hard sphere models it is possible to vary the local coordination number from closest-packed to tetrahedral, so at least in a model system it is possible to definitively examine the effect of coordination on the electronic structure. (If directional bonding is important, then in general three-body potentials will be required, necessitating a non-trivial reformulation of the integral equation theories.) Liquids are disordered; shouldn’t we see Anderson localization?² Presumably the presence of Anderson localization should shift the metal/non-metal transition; can we quantify this effect? What happens if impurities (dopants) are added? The infinite-dilution impurity case can be handled by perturbation theory, while the more general dopant case requires that the formalism be recast to handle mixtures (which is mainly a technical problem).

The next stage involves two different areas. The first is a focus on the effect of electron correlation in liquids. What level of electron correlation is “good enough” to describe the physics in a liquid? One Hubbard-like theory already exists, but no Hartree-Fock methods have yet been found that can be handled via integral equations. The second area is to treat the liquid semimetals as being composed of atoms that can hybridize. Can we develop a local-environment-specific hybridization (valence-bond) picture that efficiently builds in the electronic-structure/liquid-structure interplay? Certainly in the case of the liquid semimetals an atom with tetrahedral coordination should be hybridized quite differently than one with closest-packed coordination.

References:

1. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders, Philadelphia, 1976), p. 569.

2. Strangely, the theory of liquid electronic structure has been dormant for the last several years, despite impressive advances at the end of the 1980s and early 1990s. See R. M. Stratt, Annu. Rev. Phys. Chem. 41, 175 (1990) for a review, and also D. E. Logan, J. Chem. Phys. 94, 628 (1991) and references within.

3. F. Hensel, in Amorphous and Liquid Semiconductors, edited by W. E. Spear (University of Edinburgh, Edinburgh, 1977).

4. R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985).

5. I. Stich, R. Car, and M. Parrinello, Phys. Rev. B 44, 4262 (1991).

6. R. Virkkunen, K. Laasonen, and R. M. Nieminen, J. Phys. Condens. Matter 3, 7455 (1991).

7. Y. Waseda, The Structure of Non-Crystalline Material: Liquids and Amorphous Solids (McGraw-Hill, NY, 1980).

8. V. M. Glazov, S. N. Chizhevskaya, and N. N. Glagoleva, Liquid Semiconductors (Plenum, NY, 1969).

9. C. Li, K. Lu, Y. Want, K. Tamura, S. Hosokawa, and M. Inui, Phys. Rev. B 59, 1571 (1999).

10. R. V. Kulkarni, W. G. Aulbur, and D. Stroud, Phys. Rev. B 55, 6896 (1997).

11. J-L Lopez-Martin, E. Lomba, G. Kahl, M. D. Winn, and M. Rassinger, J. Phys. Condens. Matter 9, 3321 (1997).

12. M. D. Winn and D. E. Logan, J. Chem. Phys. 96, 4818 (1992).

13. B. Xu and R. M. Stratt, J. Chem. Phys. 91, 5613 (1987).

Estimated initial Equipment Costs (not including students and postdocs):

1. Six pentium-based computers for code development, brief production runs, and writing papers. (~ $12K)

2. Miscellaneous supplies (printer, backup hardware, maintenance, licenses, etcetera) (~ $30K)

3. Small, Beowolf parallel-computing cluster (alpha based, ~$50K for 10 nodes from www.microway.com)