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Dr Jonathan Lloyd-Williams

Theory of Condensed Matter Group
Cavendish Laboratory
19, J. J. Thomson Avenue
United Kingdom

Jonathan Lloyd-Williams

About me

I am an EPSRC Doctoral Prize Research Associate working in the Theory of Condensed Matter group of the Cavendish Laboratory at the University of Cambridge. I am interested in investigating the physical properties of materials by utilising computational methods based on the first principles of quantum mechanics. I am currently using a combination of density functional theory and the highly accurate diffusion quantum Monte Carlo method to calculate the electronic structure and atomic vibrational properties of a variety of solids.

Current research

Lattice dynamics and electron-phonon coupling calculations using nondiagonal supercells

The responses of periodic systems to perturbations characterized by a wave vector are important for probing a wide range of physical properties, such as phonon dispersion curves, electron-phonon coupling, spin fluctuations, nuclear magnetic resonance J-coupling, and many-body dispersion effects. These properties can be investigated from first principles by the calculation of total energy derivatives with respect to a particular perturbation. This can be achieved using the direct method or perturbative methods. The direct method relies on freezing a perturbation into a system and calculating the derivative of interest using a finite difference approach. Direct calculations are easily performed using standard first principles computer codes and are therefore often used for early work in a given field of research. However, only perturbations commensurate with the simulation cell can be considered exactly and the computational cost increases rapidly with system size. Perturbative methods can consider perturbations at an arbitrary wave vector using a single primitive cell and consequently have been used for the majority of calculations.

Bartomeu Monserrat and I have shown that by choosing the shape of the simulation cell appropriately, direct calculations of total energy derivatives can be performed using much smaller system sizes than previously thought possible, which reduces the computational cost by orders of magnitude. By making use of supercell matrices with nonzero off-diagonal elements, we have proved that it is possible to determine the response of a periodic system to a perturbation at a wave vector with reduced fractional coordinates (m1/n1, m2/n2, m3/n3) using a supercell containing a number of primitive cells equal to the least common multiple of n1, n2, and n3. If only diagonal supercell matrices are used, a supercell containing n1n2n3 primitive cells is required. We have applied nondiagonal supercells to perform first principles vibrational calculations using the direct method and have obtained highly converged values of the zero-point renormalization to the thermal and optical band gaps of diamond arising from electron-phonon coupling. The use of nondiagonal supercells enables the direct method to address problems that were previously only tractable using perturbative methods.

Previous research

Quantum Monte Carlo simulations of solid molecular hydrogen at extreme pressures

Hydrogen has the simplest electronic structure of any atom but its bulk properties are surprisingly complex. As hydrogen can only form a single covalent bond, it is expected to remain molecular to very high pressures. Several solid phases of hydrogen have been observed; the low-pressure phase I, which consists of a quantum crystal of rotating molecules on a hexagonal close-packed lattice, transforms at pressures of about 110 GPa to the broken symmetry phase II, in which the mean molecular orientations are ordered, and then to phase III at about 150 GPa. Phase III has been experimentally demonstrated to be thermodynamically stable beyond pressures of 300 GPa and up to temparatures of 300 K. A new phase IV was recently observed in room temperature experiments and has generated great excitement within the field. Despite many years of intensive study, the arrangements of the molecules in phases II and III remain undetermined. The structure of phase IV is also unknown.

Experimental measurements have only been able to provide limited information about the molecular orientations in phases II and III of hydrogen. The information obtained from X-ray diffraction is limited because of the weak scattering of hydrogen atoms. So far the most important experimental data have come from infra-red and Raman vibrational spectroscopy. Theoretical identification of the structures is also difficult because of the need to consider the quantum statistics of the nuclei, the large zero-point vibrational energies, and the small energy differences between competing candidate structures. By comparing Raman spectra for low-energy structures found in density functional theory (DFT) searches with experimental spectra, candidate atomic structures have been identified for each experimentally observed phase. Unfortunately, DFT predicts a metallic structure to be energetically favoured at a broad range of pressures up to 400 GPa, where it is known experimentally that hydrogen is nonmetallic.

To make progress in understanding the high pressure phase diagram of hydrogen, we have used the DMC method to calculate static lattice energies of the candidate structures at several different densities, to which we have added anharmonic vibrational energies calculated within DFT using the method recently developed by Bartomeu Monserrat and co-workers. Our calculations show that the metallic structure strongly favoured in DFT at high pressures is energetically uncompetitive and predict a phase diagram in reasonable agreement with experiment. This greatly strengthens the claim that the candidate atomic structures accurately model the experimentally observed phases.

Kinetic Monte Carlo simulations of submonolayer epitaxial growth

The Kinetic Monte Carlo (KMC) method is a stochastic technique used to simulate the time evolution of processes occurring in nature. Typically these are processes that occur at known rates which are given as inputs to the KMC algorithm. As part of my MSci project at Imperial College London working with Bartomeu Monserrat, Dimitri Vvedensky, and Andrew Zangwill, we carried out a KMC study of submonolayer epitaxial growth with a mobile intermediate cluster species, inspired by observations of graphene epitaxy on metals. We supposed that deposited atoms diffuse on the surface and collide to form four-atom clusters, or tetramers, which also diffuse on the surface, and that immobile islands are only formed upon the collision of a certain number of tetramers. Our model was able to qualitatively explain many of the findings of the graphene epitaxy study. For example, unlike all other growth scenarios, the density of atoms adsorbed on the surface at the onset of island nucleation was found to increase as a function of temperature when a large enough number of tetramers are needed to collide to form an immobile island.


  1. Lattice dynamics and electron-phonon coupling calculations using nondiagonal supercells
    J. H. Lloyd-Williams and B. Monserrat
    Physical Review B 92, 184301 (2015)

  2. Pseudopotential for the electron-electron interaction
    J. H. Lloyd-Williams, R. J. Needs, and G. J. Conduit
    Physical Review B 92, 075106 (2015)

  3. Quantum Monte Carlo study of the phase diagram of solid molecular hydrogen at extreme pressures
    N. D. Drummond, B. Monserrat, J. H. Lloyd-Williams, P. López Ríos, C. J. Pickard, and R. J. Needs
    Nature Communications 6, 7794 (2015)

  4. Epitaxial kinetics with an intermediate polyatomic species
    J. H. Lloyd-Williams, B. Monserrat, D. D. Vvedensky, and A. Zangwill
    Physical Review B 85, 161402(R) (2012)

Selected Talks

  1. Thermal expansion of graphene beyond the quasiharmonic approximation
    April 2016, Electronic Structure Discussion Group, University of Cambridge, UK

  2. Lattice dynamics and electron-phonon coupling calculations using nondiagonal supercells
    March 2016, APS March Meeting, Baltimore Convention Center, USA

  3. Quantum Monte Carlo study of the phase diagram of solid molecular hydrogen at extreme pressures
    September 2015, Psi-k Conference, Kursaal Congress Centre, Spain

  4. Direct calculation of static response functions using nondiagonal supercells
    June 2015, Electronic Structure Discussion Group, University of Cambridge, UK

  5. Quantum Monte Carlo study of the phase diagram of solid molecular hydrogen at extreme pressures
    November 2014, Electronic Structure Discussion Group, University of Cambridge, UK

  6. Density functional theory and quantum Monte Carlo calculations of solid molecular hydrogen
    July 2014, QMC in the Apuan Alps IX, The Towler Institute, Italy

  7. Quantum Monte Caro simulations of high pressure solid hydrogen
    March 2014, APS March Meeting, Colorado Convention Center, USA

  8. The search for the optimal supercell
    October 2013, Electronic Structure Discussion Group, University of Cambridge, UK

  9. A VMC study of the isotopologues of H2 and H2+
    April 2013, Electronic Structure Discussion Group, University of Cambridge, UK


Please click on the images to open a pdf version of the posters.

TEFM 2016 poster CCP9 2015 poster CECAM 2013 poster

Conferences, Schools, and Workshops

  1. APS March Meeting 2016
    March 2016, Baltimore Convention Center, USA

  2. Total Energy and Force Methods 2016
    January 2016, University of Luxembourg, Luxembourg

  3. Psi-k 2015 Conference
    September 2015, Kursaal Congress Centre, Spain

  4. CCP9 Young Researchers Event
    March 2015, University of York, UK

  5. QMC in the Apuan Alps IX
    July 2014, The Towler Institute, Italy

  6. APS March Meeting 2014
    March 2014, Colorado Convention Center, USA

  7. Psi-k/CECAM/CCP9 Biennial Graduate School in Electronic-Structure Methods
    September 2013, University of Oxford, UK

  8. QMC in the Apuan Alps VIII
    July 2013, The Towler Institute, Italy


Current collaborators

Past collaborators