Ab-initio investigation of phonon dispersion and anomalies in palladium

In recent years, palladium has proven to be a crucial component for devices ranging from nanotube field effect transistors to advanced hydrogen storage devices. In this work, I examine the phonon dispersion of fcc Pd using first principle calculations based on density functional perturbation theory. While several groups in the past have studied the acoustic properties of palladium, this is the first study to reproduce the phonon dispersion and associated anomaly with high accuracy and no adjustable parameters. In particular, I focus on the Kohn anomaly in the [110] direction.


I. INTRODUCTION
Recent interest in hydrogen storage systems and nanoscale devices has highlighted the crucial role palladium plays in a wide range of systems. Hydrogen sensors based on Pd nanowires 1 show both fast response and low power requirements. Recent experimental work indicates that Pd leads provide ohmic contacts for nanotube field effect transistors, a feature crucial for large scale device integration 2 . In addition to nanoscale applications, bulk palladium also presents interesting properties that have fascinated researchers for years. It possesses a high magnetic susceptibility and sits on the edge of magnetism. Perhaps due to this, no measurable superconductivity has been found in the system for any temperature.
Phonon scattering events play a crucial role in both superconductivity in bulk systems and transport in nanoscale interconnects. In this work, I examine the phonon dispersion of palladium from a first principles perspective. Recent advances in density functional perturbation theory (DFPT) have made it possible to examine the acoustic properties of materials at a level of accuracy previously reserved only for electronic properties 3,4 . Prior to the development of this approach, researchers were forced to use a frozen phonon technique that required large supercells or phenomenological approaches that relied on numerous fitting parameters.
Interest in the acoustic properties of palladium grew after Miiller and Brockhouse examined the phonon dispersion of palladium in detail using inelastic neutron scattering 5,6,7 .
They observed a change in the slope of the [ζζ0] transverse acoustic, TA 1 , branch of the dispersion curve around q-vector 2π a [0.35, 0.35, 0] that differed from phonon anomalies observed in other metals. The fcc lattice constant in this case is given by a. The anomaly extends over a broad range of wave vector space and also decreases rapidly with temperature. These features initially cast doubt on whether this feature was a Kohn anomaly brought about by nesting within parallel sections of the Fermi surface 8 .
For small q-vectors in metals, itinerant electrons are able to effectively screen the positive ionic charge revealed by lattice vibrations. However once the phonon wavevector, q, spans different Fermi sheets and links different electronic states, k 2 = k 1 + q, the electrons are no longer able to effectively shield the induced positive charge and the acoustic properties undergo a marked change, the Kohn anomaly 8 . Making use of Fermi surface data from augmented plane wave calculations, Miiller found transition vectors between parallel sheets of the fifth band hole surface that supported the presence of a Kohn anomaly 7 in Pd. Later calculations based on the generalized susceptibility, χ 0 (q) in Pd also indicated that nesting between sheets of the fifth band could contribute to the extended anomaly 9 . This work also predicted the presence of a weaker Kohn  Several phenomenological approaches have provided fitting routines to generate the palladium phonon dispersion. These include studies using a six parameter screened shell model 11 , a three parameter Morse potential approach 12 , and a four parameter Coulomb potential technique 13 among others. While these approaches can provide general agreement with the phonon dispersion of palladium, it should be noted that these techniques do not reproduce the phonon anomaly in the [110] direction. The large number of fitting parameters inherit in these schemes also limits the predictive ability of these approaches. One study based on a pseudopotential approach with a short range pair potential for d-d orbital interactions did show evidence for a phonon anomaly in the TA 1 branch in the [110] direction 14 . However, the predicted anomaly was shifted to smaller phonon wavevectors and showed a much more dramatic phonon softening than the measured anomaly.
Two recent works have examined acoustic properties of palladium in the framework of density functional perturbation theory. Savrasov and Savrasov looked at electron-phonon interactions for a range of materials using a full potential linear muffin tin orbital approach (FP-LMTO) 15 . Their calculations reproduce the general features of the experimental phonon dispersion of palladium, however, the phonon wavevector sampling used was too coarse to reveal phonon anomalies. A recent estimate of the superconducting transition temperature for palladium under pressure also made use of phonon dispersion curves and did not observe any phonon anomalies 16 .
In this work, I present high resolution calculations for the phonon dispersion of fcc palladium. I compare direct calculations at q-vectors along high symmetry lines with results based on interatomic force constants. In particular, I focus on the phonon anomaly in the [110] direction and show that DFPT is able to reproduce this feature with high fidelity.
Since this anomaly is due to phonon-electron interactions, I also determine the location of the Kohn anomaly based on Fermi surface analysis and the two different techniques show good agreement. I have also examined the phonon dispersion along the [111] direction and I do not find evidence for the phonon anomaly previously predicted by Freeman et. al. 9 .

II. DENSITY FUNCTIONAL PERTURBATION THEORY
For a periodic lattice of atoms, the only portions of the crystal Hamiltonian that depend on the ion positions are the ion-electron interaction terms, V R (r), and the ion-ion interaction term, E ion (R). The force on a given atom consists of a component that is responding to the surrounding electron charge density, n(r), and another term that accounts for ion-ion interactions. The interatomic force constants used to determine phonon frequencies in the system are found by differentiating the force, F I on a given ion, I, by the change in position of ion J.
The interatomic force constant matrix (IFC) above consists of seperate electronic and ionic contributions. The ionic contribution can be related to an Ewald sum and calculated directly 17 .
There are two factors that are critical for the determination of the electronic contributions to the IFC. The first is the ground state electron charge density, n R (r). The second is the linear response of the ground state electron density to a change in the ion geometry.
These quantities can be calculated directly within the density functional framework without resorting to fitting data from experiment.
While first principle calculations readily determine the equilibrium electron charge density, n R (r), some additional effort is required to determine ∂n R (r)/∂R I and the corresponding IFC. However, the computational effort required is on the same order as the standard ground state energy calculation. Interatomic force constants for periodic structures can be determined through the use of density functional perturbation theory 3,4 . In this approach, the Kohn-Sham equations for the charge density, self-consistent potential and orbitals are linearized with respect to changes in wave function, density, and potential variations. For phonon calculations, the perturbation, ∆V ion , is periodic with a wave vector q. This perturbation results in a corresponding change in the electron charge density, ∆n(r). Since the perturbation generated by the phonon is periodic with respect to the crystal lattice, we can  20 . In both sets, the effect of including semi-core d states was also considered. I found that all GGA pseudopotentials predicted a magnetic ground state for fcc Pd. In addition, LDA pseudopotentials that did not take into account the semi-core d state predicted a magnetic ground state as well. Bulk palladium is not ferromagnetic, although previous works have indicated a small expansion of the crystal can induced a finite magnetic moment 21 . The LDA pseudopotential that includes the semi-core d state provided the physical paramagnetic palladium ground state and is used throughout the remainder of the study. The inability for GGA Pd pseudopotentials to provide the correct non-magnetic ground state has also been recently found by other groups using different first principle approaches 22,23 . It is also interesting to note that a recent work examining the phonon dispersion in fcc palladium found good agreement with LDA pseudopotentials, but poor agreement with GGA pseudopotentials 16 . Since the Pd GGA pseudopotential overestimates the equilibrium unit cell volume, this leads to an overall shift in the predicted phonon dispersion to lower frequencies.
B It should be stressed that the phonon dispersion curves from both the direct calculations and interatomic force constants are calculated without any adjustable parameters and make no reference to experimental data. There are two clear sources for differences between the measured and calculated phonon frequencies. The first is due to the fact that experimental studies were performed at 120 K, while first principle calculations assume zero temperature.
Thermal expansion of the unit cell will act to shift the phonon frequencies lower. In addition, the local density functional approximation (LDA) has a tendency to overbind and predict slightly smaller lattice constants than those observed experimentally. Using LDA, the equilibrium lattice constant for fcc Pd was determined to be 3.88Å, while the measured lattice constant is slightly larger, 3.89Å. The smaller predicted lattice constant will lead to a more compressed unit cell and a general shift to higher phonon frequencies. This trend of overestimating the phonon frequency is greatest in the longitudinal acoustic branch.
While there are some slight differences between the current work and previous experi- The Fermi surface for fcc Pd was extracted from the calculation as well. There are three bands that contribute at the Fermi energy ( Fig. 4(a)). The fourth band barely overlaps with the Fermi energy and only exhibits small cusps at the center of the [100] faces. The sixth band is centered in the Brillouin zone and although faceted, has high s character and is similar to a free electron Fermi surface. The Fermi surface for the d-like 5th band in fcc palladium is shown in Figure 4

IV. CONCLUSION
In this work, I have considered the lattice dynamics of fcc palladium based on density functional perturbation theory. Agreement between direct phonon calculations along symmetry lines with current experimental data for the full phonon spectrum of palladium is good. In particular, I was able to reproduce the extended phonon anomaly in the [110] direction that had been observed experimentally. In addition, I found no evidence for phonon anomalies in other branches of the phonon dispersion. I also examined the phonon dispersion derived from interatomic force constants and looked at the influence of q-vector sampling.
The interatomic force constant approach with a 8x8x8 q-vector grid was able to provide good agreement for general features of the palladium phonon dispersion.

Acknowledgments
Calculations were performed on the Intel Cluster at the Cornell Nanoscale Facility which is part of the National Nanotechnology Infrastructure Network (NNIN) funded by the National Science Foundation. Fermi surface images were generated using XCrySDens software package 27 . I also wish to thank Eyvaz Isaev for providing the Fermi surface conversion utilities for Quantum Espresso.